This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
since 9 is connected, A = 9 and u = u. Once I is proved, it follows that the additional hypothesis under which we proved III was superfluous. In order to prove II assume that u achieves a positive maximum at a point x0 in W. Then au < 0 in a neighborhood of this point and in this neighborhood Mu = Lu  au > 0. By I, u = u(x0) in a neighborhood of x9. Thus the set of points at which u = u(xo) is open. Since it is closed by continuity, it must coincide with the whole domain 9.
For the proof of IV, assume that k is a point on I with = max u = u(.x) > 0. Then Mu = Lu  au > 0 near z and, by III, either du/d1V > 0 at X or u  µ at some interior points close to X. In the latter case u is constant by II. 2.3.
Applications to the Dirichlet Problem
We shall now discuss some applications of the maximum principle to boundary value problems. Consider first the Dirichlet problem u = O on (3) Lu =f in Vt, where f, 0 are given bounded functions.
THE MAXIMUM PRINCIPLE THEOREM.
153
If a < 0 then the solution u of the Dirichlet problem (3)
satisfies the inequality
Jul < max 101 + (e"  1) max If I
(4)
where a = (11m) [K { (K2 + 4m)112] and d is the diameter of 9.
The interesting thing about this inequality is that it does not depend upon the size of a, or on the continuity properties of the coefficients, or on the shape of the domain.
An immediate corollary of this estimate is the uniqueness theorem for the Dirichlet problem. If f = 0 = 0, a < 0 and (3) holds, then u = 0. Of course, this also follows directly from II. In order to prove the estimate we may assume, without loss of

generality, that 9 lies in the strip 0 < x1 < d. Set g(x) = e' eu'. In the strip considered, we have that e"d  1 > g > 0 and Lg = (a11a2 a1x)e'zl + ag < mat ; Kx = 1. Now set

h =maxI0I +g(x)maxIfI.Then Lh =maxIf(Lg+amaxI0l max If I and h > max 101. Hence v = u  h < 0 on 9 and Lv=f  Lh >f } maxIfI >0 in W. By II, v :!5: 0 in 1.
Similarly, u ! h. Hence Jul < max 101  max If I g(x) which implies the assertion. The proof shows that the theorem can be considerably strengthened.
If a is not nonpositive, we cannot expect in general that the Dirichlet problem will have a unique solution. Nevertheless, the
uniqueness theorem can be proved and an estimate can be obtained if the domain is sufficiently small. THEOREM. Assume that a < k where k is a positive number, and that the diameter d of <9 is so small that e'"1  1 < 1 /k. Then every
solution of problem (3) satisfies the inequality (5)
IuI
c maw ICI ; (e«a  1) max If 1 k(e""1)
This implies that under the hypotheses of the theorem the Dirichlet problem has at most one solution.
For the proof of the theorem we set a = (a + Ial)/2, a _ (a  Ial)/2 and observe that the function u may be considered as
a solution of the Dirichlet problem for the equation Mu  au = ,
,
f =f  a+u. Since a < 0 and 0 < a" < k we have that
L. BERS AND M. SCHECHTER
154
max If I < max If I + k max Jul and (4) yields max Jul < max 101 +
(e«a
 1) (max if I + k max Jul)
which implies the assertion. For a < 0 the maximum principle can also be used in order to prove the existence of a solution of the Dirichlet problem under the hypothesis that we know how to solve this problem for "nice" domains, say for small discs (Perron's method of upper and lower functions). But we shall not discuss this here.
2.4.
Application to the Generalized Neumann Problem
As another application we consider the oblique derivative problem. We assume that the boundary of 9 is sufficiently smooth and we are looking for a solution. of the differential equation Lit =f satisfying the boundary condition on I
Here a/aa indicates differentiation in a direction depending on the point considered. We require that this direction should be nowhere tangent to the boundary surface ik. A solution is assumed to be of class Cl in c. THEOREM.
If a < 0 and ul and u2 are two solutions of the oblique
derivative problem formulated above, then the difference u1
 u2 is a
constant.
Set u = u1  u2. Then Lu = 0 in V, au f as = 0 on 9. Since u is continuous in 9 it achieves its maximum at some point £. Proof.
If u is not a constant, we may assume that max u > 0. By II, z
must be a boundary point. Near this point Mu = au > 0 so that the normal derivative of u must be positive at f by III. But
all tangential derivatives vanish at X. This implies that the prescribed derivative au/aa = 0 at x, which is impossible.
THE MAXIMUM PRINCIPLE
155
2.5. Solution of the Dirichlet Problem by Finite Differences
As the next application of the maximum principle, we shall discuss the effective numerical computation of a solution of the Dirichlet problem (3) by the method of finite differences. For the sake of simplicity, we restrict ourselves to two dimensions and to a differential equation of the form
Lu = Du + alu, + a2u,, + au =f
(6)
The functions al, a2 and a are assumed to satisfy the conditions
aS0 We approximate the differential operator L by the finite $a11 + 1a21 < K,
difference operator
Lhu=Mhu+au
(7)
= h2[u(x + h, y) + u(x  h,y) { u(x,y + h) + u(x, y  h)  4u(x,y)] u(x + h,y) u(x h,y)

+ a1(xy)
2h
u x, y
u x, y + a2 (xy)

+ a (xy) u (x,y)
2h
the mesh width h being a small positive number. Note that Lhu
Lu as h
0 for every C2 function u.
In order to formulate a boundary value problem for this difference equation, we introduce the following notations. The points
(x + h, y), (x, y + h), (x  h, y), (x, y  h) will be called the hneighbors of PO = (x, y) and will be denoted by P01, P021 P03, P. Py situated in 9, A lattice domain'h is a set of points Pi, P2,
....
having coordinates which are integral multiples of h, and such that for every point in the set its hneighbors belong to K We also require that given any two points Qo and Q00 in 'h there should exist points Qo = Q1, Q2, ... , Q, = Q00 in Wh with Qj
an hneighbor of Q,_1. Neighbors of points of Fh which are not
themselves points of Th form the boundaryh of !Yh; these points will be denoted by PV+1, PV+21 ... , P' ,sr. The union of
L. BERS AND M. SCHECHTER
156
yh
and 'h is the "closure" 1h of !Ye,,. We assume that h is so
small that a lattice domain 65h exists and that
hK < 0 < 1
(8)
For the difference equation we now pose the following Dirichlet problem (9)
Lhu
=fatP1,P2,...,P.V;
u =0 atP.,+1,.
.
, PV+.11
The solution u is of course to be defined only at the N i M points P,.... , Pv h,,, or as we say, on the lattice 7h.
We claim that the following maximum principle holds for functions u defined on the lattice domain h. If Mu > 0 in 1,
then either u is a constant or it achieves its maximum on the boundary The proof is trivial. For a point Pi in F,, the condition 0 reads Mhu
u(Pi) <
4
j
1.,,u(P,,) 1
.i2 = 3
1 (1 41`
417a ) 1
t
L
 Ilal 2
where we denoted by a1, a2 the values of a1(x,y), a2(x,),) at the point P,. The assertion now follows by observing that the ).,, are positive and their sum is 1. Thus if u(P,) = max u, then u = max u at all hneighbors of P at all hneighbors of the hneighbors, etc. Hence u  max u. From this maximum principle corresponding to Theorem I
above we conclude, just as before in the case of differential equations, that a function defined on the lattice domain and satisfying the inequality Lhu > 0 in,, cannot have an interior positive maximum unless it is a constant. Furthermore, the same reasoning which led to the estimate (4) leads now to the following estimate for the solution of the Dirichlet problem (9) : (10)
Jul < max 101 ; C max If I
THE MAXIMUM PRINCIPLE
157
where C depends only on K, the domain T, and 0. In fact, we may use the same comparison function as before since we have that Lh(ead  exz) _ a2 e,,x(sinh (ah/2))2 ah/2
blot e"x sinh (ah) + a(e"d  e"s) /2 )12l(1
(sin
 ot2 a
ah/2
J
 K ah coth (ah/2) J <  1
if a > 0 is large enough, provided that 9 lies in the strip 0 <
x
system, and we have just found that the homogeneous system possesses only the trivial solution. Therefore, the Dirichlet problem for the difference equation is always uniquely solvable.
Next we want to show that the solution of the difference problem approximates the solution of the differential problem as the mesh length h
tends to zero. To do this we shall assume, merely for the sake of simplicity, that for a sequence of values of h tending to 0, all boundary points of W. lie on the boundary 1.
We shall assume also that the solution u of the Dirichlet problem (3) for the differential equation Lu =f exists and has bounded derivatives of order three. An easy application of Taylor's theorem shows that for any C3 function u
JLhu  Lul < Ch the constant depending on the maximum modulus of the third derivatives of u. (For a C4 function Lhu
 Lu =
0(h2) .) Now the
estimate (10) shows that every function U defined on the lattice domain 1h satisfies the inequality
maxIUt <maxIUj =CmaxILhUj yA
'A
A
L. BERS AND M. SCHECHTER
158
Applying this to the difference u u(h), u"' being the solution of (9) we obtain the uniform estimate
u  u(h) = 0(h) which shows that the solution of the difference equation converges to that of the differential equation. It should be noted that the assumption made on the shape of the domain is unessential. As a matter of fact, without assuming anything about the boundedness of the third derivatives of the solution u of (3) on the closure !k, but merely the continuity of
these derivatives in 9, one could say that the solution of the Dirichlet problem for the differential equation can be approximated by properly chosen solutions u(h) of the difference equation
on every compact subset of W. As a matter of fact, this same method works for uniformly elliptic differential equations of the
form (1) and even for semilinear equations of the form Mu =
f
(Bers [2]). It is only necessary to be able to approxi
mate the differential operator L by a finite difference operator L. for which one can prove the maximum principle. This is always possible, though it might happen that one needs a difference operator involving many more neighboring points than the simplest difference approximation used above (Motzkin and
Wasow [1]). Also, the restriction to two dimensions is completely unessential. Finally, a refinement of the argument would show that under proper conditions the first and second difference quotients of the
solutions of the difference equation converge to the first and second derivatives of the solutions of the differential equation. As a matter of fact, the argument could be reworked so as to give an
existence proof of the Dirichlet problem for the differential equation. 2.6. Solution of the Difference Equation by Iterations
The problem of solving the Dirichlet problem for the difference equation numerically, say on an automatic computing machine has been the subject of intensive investigation for which we refer to the literature. We shall only show how the maximum principle
THE MAXIMUM PRINCIPLE
159
can be used in order to prove the convergence of a simple iteration scheme (Liebmann's method).
It is now slightly more convenient to consider the Dirichlet problem for the homogeneous difference equation :
L,,u=0inIi.,
u=0on4,,
Liebmann's procedure is as follows. We assume some trial values
for the unknown quantities u(P1), . uo(Pi),
. . ,
u(P,;) and call them
... , ua(P.,). We order the points of the lattice domain in
such a way that P1 has as an hneighbor a point on the boundary 4r,,, P2 has as one of its hneighbors Pi, and so on. Now we correct the trial value u,,(P1) so that the difference equation should be satisfied at this point. We then correct, in a similar fashion, the trial value uo(P2), making use of the new value u1(P1). After having corrected the values at all points P1, ... , Pv we obtain the next approximation u1(PI), ... , u1(PN). We now repeat the procedure in order to obtain approximations u2, us, We want to show, following Lowner (cf. Frankv. Mises [1]), that v = 1, 2, ... , N lim u,(P,) = u(P,),
....
5. ao
where u is the solution of the boundary value problem (9). To do this we observe that the kth approximation uk depends linearly
on the (k  1)st approximation and on the given boundary values 0; thus there exist matrices (a,,,), (#,.,) such that N
M+N
uk(P,) _14si I a,Puk1(PH) ,,=,v+1 + I f,,,4(P,,) It is easy to see that every correction consists in replacing the value uk_1(P,) by a linear combination, with positive coefficients, of the values of uk_1 at neighboring points. Hence the elements a,µ > 0. If we had luck enough to choose as our first approximation the exact solution, the corrections would be unnecessary. Hence and therefore
u(P,) = I a"Hu(P,,) + I #,,,O(P,.)
max, I uk(P,)  u(P,) I = Max, 11 a,,,(uk1(P,.)  u(P,.)) I max
a,,, Iuk1(P,,)  u(P,,) I <_ 0 max Iuk1(P,)
 u(P,) I
L. BERS AND M. SCHECHTER
160
where 0 = max,, E a,,,,. Now assume that we choose as our first approximation the function u = 1 and that the given boundary values are ¢ 0. Then each correction consists of diminishing
the trial value; and the first approximation would consist of positive values less than 1. But the value of the first approximation at P. is now Z OCR,,. Thus 0 < 1, and since
max tux  uj < 0 max IUk_1  uI < ... < 0' max Iuo  uJ we see that the Liebmann method converges.
The maximumprinciple approach to the method of finite differences is limited to secondorder equations. An alternative approach (CourantFriedrichsLewy [1] and others) uses L2 estimates and works also for equations of higher order. 2.7.
A Maximum Principle for Gradients
First derivatives of solutions of linear elliptic equations do not, in general, obey the maximum principle. An exception is a second
order equation in the plane of the form (11)
A(x,y)uzx + 2B(x,y)ux , C(x,y)uvv = 0
(AC  B2 > 0). If u is a solution defined in a domain 1, then ux and u cannot achieve their maxima at interior points, unless they are constant.
We shall prove this in Chapter 7. At the moment we note that the assertion is obvious if A, B, C are of class C1 and u of class C3. Indeed, dividing the equation by C (or by A) and differentiating with respect to x (or toy) we see that u,, (or satisfies a secondorder linear elliptic equation. For an equation of the form (12)
"
Au
i1
a,
au
ax;
} au = 0
with bounded coefficients we can state the following THEOREM.
Let K be a bound for the moduli of the coefficients in (12).
There exist positive numbers Ro and k depending only on K and n such that for every solution u of (12) defined in a convex domain A of diameter
THE MAXIMUM PRINCIPLE
161
R < Ro, of class C2 on the boundary I' of A, and vanishing somewhere in A,
sup, (grad uj < k maxr (grad uI Set ,u = supA Igrad uI, v = maxr (grad uI. Since u = 0 somewhere in Jul < Rµ, and by (12), assuming that R < 1, Proof.
zul < K(Vn r R),u Set v (x)
=  I Au(y) dy
v(x) =
2
(n > 2)
_2
1
J' Au (y) log
x
dy
(n = 2)
yI
where uw,, is the area of the unit sphere in nspace. Computing the gradient of v by differentiating under the integral sign we obtain easily the estimate sup, Igrad vi < cKR,u where c depends only on n. The function v(x) is of class C1 everywhere, and by Green's third identity, fi(x) = u(x)  v(x) is harmonic in A. Now, the derivatives of 0 are harmonic and obey the maximum principle. Hence
sup, (grad I < maxr (grad 01 < v + cKR4u so that
it < sup, Igrad 01 + cKRµ < v ! 2cKRy If R < Ro < 1 /2cK, we see that 1u < kv, with k = 1/(l  2cKRo). The convexity of A was used only in asserting that max., Jul < R max, (grad ul ; it would suffice to assume 0 starshaped with
respect to the point where u vanishes. For nonconvex ., with sufficiently smooth boundary the theorem remains valid, with k depending on A. No convexity is needed if a = 0. In this case the condition that u vanish somewhere in A is also superfluous: it can be achieved by adding a constant to u. The same method yields a maximum principle for an elliptic system in the plane Ox  tVv = x111  a12V (13)
= x210
x22tV
L. BERS AND M. SCHECHTER
162
with bounded a;;. This is not surprising since for the special case a21 = a22 = 0, system (13) shows that 4 = u,,, tp = uv where u is a solution of E u + a11um + a12u = 0 Setting 0 + i v = w we may write (13) in the complex form
(14)
w,, =aw±bw
(15)
where a and b are bounded complexvalued functions. This equation implies, of course, the inequality Iw2I < K IwI
(16)
and for such an inequality we have the following THEOREM.
Let w(z) be a function of class C1 in a domain A of diameter
R < R0, satisfying inequality (16), and assume that w is continuous on the boundary 1' of A. Then (17)
supA IwI < k maxi, IwI
Ro > 0 and k > 0 being constants depending only on K.
The proof consists of applying the maximum modulus theorem to the holomorphic function w(z)  r(z), z e A, where (18)
r(z)
iff
w,( z) dz dry
+ i?l
0
and noticing that sup_A Irl < 2R sup,, I wtl < 2RK sup,, IwI. 2.8.
Carleman's Unique Continuation Theorem
The maximum principle just stated implies a celebrated theorem of Carleman. THEOREM. A function w(z) of class C1 which satisfies inequality (16) in a domain V vanishes identically if it has in 9 a zero of infinite order.
Assume that w(z) = 0 (Izl`), z ) 0, for all N > 0. It suffices to show that w(z) = 0 for Izl < R, R > 0 and small Proof.
THE MAXIMUM PRINCIPLE
163
enough. The functions w(z) n = 1 , 2, ... , vanish at z = 0 and satisfy (16). Applying (17) to the domain 0 < I zI < R, we see that max I z"w(z) I< kR" max I w(z) I Izj
for n = 1, 2,
Izl=R
.... This is impossible unless w = 0.
In view of the connection between system (13) and equation (14), Carleman's theorem implies at once that solutions of (14) have the weak unique continuation property. In order to obtain
the strong unique continuation property we must be able to conclude from (14) and the relation u = 0 (Izl `), for all N, that U. and u also vanish of infinite order. This is indeed possible. The strong unique continuation theorem for the equation (19)
Au+au,, +buy±cu=0
is obtained by noting that this equation can be reduced to the form (14) by introducing the new unknown function U = u/uo, uo being some positive solution of (19). Finally the general linear
elliptic secondorder equation in the plane is reducible to the form (19) under certain smoothness conditions on the coefliciens. A unique continuation proof not tied to our smoothness assumptions will be sketched later (cf. Chapter 6).
CHAPTER 3
Hilbert Space Approach. Periodic Solutions 3.1.
Periodic Solutions
The Hilbert space approach to boundary value problems is the end result of a development which goes back to Riemann. As is
well known, Riemann asserted the existence of a harmonic function with prescribed boundary values on the basis of the Dirichlet principle, that is, by characterizing harmonic functions as extremals of a variational problem. The criticism by Weicrstrass led to the abandonment of this method of attack, but around the turn of the century Hilbert showed that the Dirichlet principle can be used for rigorous existence proofs. The direct methods of
calculus of variations which developed from this result are described in several books (Weyl [1], CourantHilbert [1], Courant [11) and we shall not discuss them here. It was realized later that
the natural framework for the Dirichlet principle approach to boundary value problems is the theory of Hilbert space (Zaremba [ 1,2], Nikodym [1], Friedrichs [2], Sobolcv [2] and others) and that Hilbert space methods are not limited to secondorder equations or to equations arising from variational problems (Visik [1],
Garding [1], Browder [1], Friedrichs [2], Ladyzenskaya [1], Nirenberg [1], LaxMilgram [1], and others). As a matter of fact this approach to boundary value problems uses no deep results from the theory of Hilbert space proper. The main tools (described fully in Appendix I to this chapter) are the projection theorem and the representation theorem. Note that the representation theorem is an easy consequence of the projection theorem and the latter result is proved by minimizing the distance between an clement in the space and a closed linear subspace. In this sense, calculus of variations is still present in the Hilbert space approach. On the other hand, one must also use some deeper results: the RieszSchaudcr theory of linear equations in a Banach space (see 164
HILBERT SPACE APPROACH, I
165
Appendix II to this chapter). This is an abstract analogue of the Fredholm theory of integral equations. It is worth noting that the Fredholm theory itself was developed mainly for solving the Dirichlet problem without using calculus of variations. We shall describe the Hilbert space methods first for a somewhat
artificially oversimplified case. Instead of treating a boundary value problem we shall be looking for a periodic solution of an elliptic differential equation. In the periodic case certain technical difficulties disappear (as was observed by Lax [1]) and a partic
ularly elegant treatment is possible. This treatment also makes essential use of the theory of distributions. More precisely, we consider in this chapter an elliptic differential operator of order m, L = a p(x)D" and assume that the Irl»:
coefficients of L are periodic in all variables. Without loss of generality we may set all periods equal to 27r. We also assume, primarily for the sake of simplicity, that the coefficients of L are
infinitely differentiable. From now on all functions will be required to have period 27r in each variable, and all integrals will be performed over a period cube: 0 < x, < 2ir, i = 1, 2,
. . .
, n.
The Hilbert Spaces Ht
3.2.
First we consider realvalued trigonometric polynomials, that is, finite sums of the form u = u(x) = I ocle`f . Here e = (lr, ... , are ntuples of integers (positive, negative, or zero), I  x = fix, + . . . + and the (complexvalued) coefficients ap are
subject to the condition a_t = al. For t = 0, + 1, ±2.... we define the scalar product of u =
age'f'r and v =
ifte'f by
the formula (1)
(u,v)t = (2ir)" I (1 + f. t)t at#f
2(where
I = B; ... + f) . The axioms of a (real) scalar product are obviously satisfied and we note that }
(2)
(3)
(u,v)o = fu(x)v(x) dx
Oct = (u,e'fs) o
L. BERS AND M. SCHECHTER
166
The product (1) induces the norm II
II
(1 + t
Ilullt = (u,u) t = (27T)"
(4)
t,
l) t l atl2
We note the Schwarz inequality I (u,U)sl S (lulls IlUlls
(5)
and the generalization (sometimes called the Schwartz inequality) (6)
l (u,U)sI C llulls+t IlUllst
For u = v this yields (7)
IIuIIB < llulls+t llulls_t
Also, (8)
Ilui18 S IIuII t
for s < t
From the inequality (1 + e 1)8 < 8(1 + e. 1) '1 + e(stp)1(t,s) (1 + `. f)t:
we obtain (9)
E(3csu(t,s)Ilullts
IIuIIB < ellullt,
for ti >S > t2,
e>0
For every partial derivative DP, (10)
DP
ate't'x = I (if).
where, according to our conventions, (ie)P = (i11)Pl(i4)P2
...
(i(,)P.
Hence (11)
IIDPull: < Ilullt+Inl
and (12)
Ilullt < (const. depending on t) I IIDPullo
Now let K denote the elliptic differential operator K = 1  A.
Then K I ate't'Z = (13)
KtI
(1 + t l) ate't'Z, so that for t = 0, ± 1, ateit'
= I (1 + !' I)
tatefttz
HILBERT SPACE APPROACH, I
167
Clearly, (Ktu,v) 8 = (u,K`v) d = (u,v) a+t
(14) (15)
IIKtull8 =
Ilulle+2t
Completing the space of trigonometric polynomials with respect
to the norm Il t we obtain the (separable) Hilbert space H. The elements of Ht may be represented as (infinite) trigonometric octe't' series (with a_g = 61) for which the norm (4) is finite. II
Clearly
Ht cH,
(16)
We set H. = n Ht, H
fort>s
. = U Ht.
All relations developed above remain valid for elements of H. In particular: D9, as defined by (10), is a bounded linear transformation of Ht into Ht_,,,, and the scalar product (u,v), may be formed if u c H,,,, v c H.,,. 3.3.
Structure of the Spaces Ht
A formal trigonometric series E age"' is the Fourier series of a periodic squareintegrable function if and only if E I atl2 converges (RieszFischer theorem). Hence there is a onetoone correspondence between such functions and elements of Ho. We state this, somewhat imprecisely, as LEMMA 1.
Ho is the space of periodic L2 functions.
Noting (11) we also have LEMMA 2.
For t > 0, Ht is the space of periodic functions having
generalized L2 derivatives up to the order t.
In particular, a periodic function of class Ct belongs to H. A partial converse of this statement is also true. LEMMA 3.
(Sobolev [1]). If u(x) E H, and t > [n/2] + k + 1,
then u is of class Ck, and (17)
max IDDul < const. Ilullt
for IpI < k
L. BERS AND M. SCHECHTER
168
It will suffice to prove this for k = 0 (cf. (10)). Now if u = E afe'f r c Ht, t > [n/2] + 1, then
Ixfl)I
so
max Jul < E lxfl < (E (1 + 1. LEMMA 4.
1)t)112
IluII1.
H., is the space of Cc functions.
This follows at once from Lemmas 2 and 3. The spaces H, and H_ t are dual via the product ( , ) o. More precisely: if v c H t, then cw(u) _ (u,v)o is a bounded linear functional5 on Ht, and Il v II _t = II of ll, that is, LEMMA 5.
IIvII_t = sup (u,v)o,
(18)
Ilullt = 1
Conversely, every bounded linear functional on H, is of this form, v being uniquely determined.
Proof.
If co(u) is given, the representation theorem implies
the existence of a uniquely determined element i E H, such that (,,(u) = (u,b), and 11(011 = IIUIit Set v = Ktu. Then v c H_,, Ilvll_t = lI'llt
and (u,v)o = (u,v)t = cv(u). If v1 e H_, and co(u)
(u,v1)0, then
(u, v  v1)0 = 0 for u E Ht, and for u = K'(v  v1) we have that Ilv  v1Ib_t = 0 or v = v1. If v e H_, is given, (u,v)o is a bounded linear functional on H, by (16). A periodic distribution T is a linear functional defined on C.,
periodic functions 0, such that T[s6k]  0 if D"Ok , 0, k , oo, uniformly, for every p. By Lemmas 3, 4, and 5 this means:
T[ck] 0ifII¢kll,*0, k oo, for all t>0. IfueH_t, t>0,
we may define the linear functional T[¢] = (u,¢)0. Since C. functions are dense in H, we have LEMMA 6.
H_t (t = 1, 2,
. .
.)
is
the space of those periodic oo. 0, k
distributions T for which T [Ok]  0 if IlOkllt
We also have LEMMA 7.
(L. Schwartz [1]). H_ x is the space of all periodic
distributions. 5 See Appendix I to this chapter.
HILBERT SPACE APPROACH, I
169
Suppose that T is a distribution not in H_,,: T[¢] 0 (u,O)o for all u e H_t, t = 1, 2, .... By Lemma 6 there exist C., functions 0k, k = 1, 2, ... , with I T[tk] I > k IIOkIIk Set zVk = Ok/k Il0kilk For every fixed s = 1, 2, ... , Ilwkll,  0, since Proof.
Iiv'kll, < 1/k for k > s. But I ThA] I > 1, which is impossible. For periodic distributions we can define differentiation (which coincides with the operation defined by (10)) and multiplication by a periodic Cx function. They are similar to the corresponding definitions for general distributions given in Section 1.3. If 0 is a C,,, function and u e H, then Ou E Ht and
LEMMA 8.
< (const depending on 0) II u II r Proof. It suffices to verify (19) for u E Ham. If t >_ 0 the
(19)
II Ou II i
inequality follows at once from (11,12). If t = s < 0 we have by (18) and (6) that I* II, = sup {(0u,v)0/ Ilv II,} = sup {(u,iv)o/ Ilv II,} Ilull , II/v1I,/Ilvll, < const. Ilull ,
As a corollary we obtain LEMMA 9. If L is a (not necessarily elliptic) partial differential operator of order m with periodic C,, coefficients and u E Ht, then
Lu e H,. and IILu li tm < const. Ilu ll,
(20)
We conclude by a lemma which shows that the embedding of Ht into H87 s < t, is a completely continuous mapping. LEMMA 10.
(Sobolev [3]). For s < I every bounded set in H,
is conditionally compact in H3. Proof. Ilu(')11
Let {0) = {E a` >e'f'z} be a bounded sequence in Ht,
M. We must show that a subsequence is a Cauchy
sequence in Ha. Since I at ) I < M(1 , f i) tl2 we may assume, selecting subsequences if need be, and using the diagonal procedure,
that for every t the sequence {m(P) converges. For every integer U) N > 0 set u(') = u(j)N + VU) N , where uN = E x( Pe'f z, the sum
mation being extended only over those l for which t° l < N2. Clearly Iluv)
 u(' ) II3 will be arbitrarily small, for every fixed N,
L. BERS AND M. SCHECHTER
170
if j and k are large enough. On the other hand Il vY'  v1; II2 S const. (1 + N2)stM2 is arbitrarily small, for all j and k, if N is large enough. Hence II u(')
 ue4l , i 0 for j, k + oo.
3.4.
Basic Inequalities
We consider now our elliptic operator L = E a,(x)D' and establish the basic a priori estimates. THEOREM
1.
(Garding's Inequality). For every periodic C.
function u (21) (u,Lu)0 . cl Iluiln,f2  C2 Ilullo where ci and c2 are positive constants depending on L.
Inequality (21) holds also for u e Hm/2 Indeed, if u c Hmi2i then Lu E H mi2, so that the scalar product (u,Lu) 0 makes sense, and (21) may be proved by a limiting process. COROLLARY.
THEOREM 2.
For e v e r y t = 0, ± 1 , ±2,
... ,
and every periodic
C,,, function u
(22)
Ilullt < c3(t) IILu + Aull:m
for A > A
where c3(t) depends on t and L and A > 0 on L only.
Again we have a COROLLARY.
Inequality (22) holds also for u e H.
Proof of Theorem 1.
We remark first that by hypothesis
(1)m12 I
c>0
c IEIm,
IpI =m
for every vector . Case (a). L has constant coefficients and only terms of highest degree m. Then L = I aDDD and for u = E a`e'e'z 1P1=m
(u)Lu)o = ( aeetlz
G
(I
Ip1 m
(1)m12Q(I) Iafi2 > C c
[1 + (t'
C' IIUI12
where c' = c sup {(1
2rm)(1
av(ie)v)a(eatz)o
t')m/2] (xfl2
C IIullo
+
52)m/2}.
(l. 1)m12 I0CfI2
C
Ixll2
HILBERT SPACE APPROACH, I
171
Case (b). L has only terms of highest degree m and is of the form
L = Lo + L1 where Lo has constant coefficients while Ll = b,(x)D9 with max lb,(x) I
n
W =M
(u,Lou) o
(23)
z
sufficiently small. By (a)
c' Il u lI1/2  c
Il u ll o
Integrating by parts and noting that boundary terms disappear because of periodicity we have that (u)Llu)o
=f(
where
Ip
Imbp(x)u(x)D3'u(x)} dx = Ii + I2
Il = l bDDr'uD".u dx summed over Ipl = m, IP'I = Ip"I = m/2, $b3,qD2UD1U dx
I2 =
summed over Ipl + IQI < m  1, Ip1 < m/2, I QI < m/2, the
being C. functions. Noting the inequalities in Section 3.2 we conclude that I1I = I 1121
(b,,D'° u, D?u) o l < 71 const.
II U II m f2
= Ej (b,,Dpu, DPu)0I < const. IIu U IIu II(m/2)1 COIISt. + (em/2) Ilullo] IIuI1m12{EIIulIm/2
< e const. IIu II
For small a and (24)
n12
±
(em/2)
const. IIu III,
e>0
these inequalities, combined with (23), yield
(u, Lou + Liu) > ci
11'
t
 cl
(ci > 0)
1 1u1 1u
Case (c). L = Lo + Li + L2 where Lo and Ll are as in Case (b)
and L2 =
aD(x)DP. By integration by parts and by the Ipl <m
argument used before (u,L2u)o = 1J a,qD3uD°u dx
IPI + IQI < m,
I pI < m/2,
IQI < m/2
L. BERS AND M. SCHECHTER
172
so that (25)
(u,L.2u)01
I
const. IIu ll,r,/2 Ilu II (rn/2)1
>0 For small a this together with (24) yields Garding's inequality. General Case. For a sufficiently small n > 0 we construct e const. Ilu
(e"'/2) const. Ilu 112,
periodic C. functions aq(x), (u2(x), . . properties. (i) On the support of
.
,
(o).\(x) having the following
each w, the oscillation of IpI = m, of L is less than 71.
each leading coefficient (ii) E (i)1(X)2  1. By Case (c) (26)
((u,u, Lo);u) o > pos. const. II oo;u Il ,,,2  const.
II (,),u II 2
but (u,Lu) 0
(27)
f
= (I (1 )uLu dx =
((o,u, L(o;u)o ± R
R = 1 f(f))u {L(oju  (u,Lu) dx Integration by parts yields
R=J
dx,
IpI + 191 < m,
with some fixed C,, functions
141 < m/2
and as before (em/2)
IRI < e const. Ilu IIn,/l +
(28)
I pI < m/2,
const. Ilu Il0
Clearly II(,),u Ilo < Ilu Ilo
(29)
and it is not difficult to verify, using for instance (11,12), that (30)
Il(,.),u II,,,/., > pos. const.
II U
II71t/2  const. Ilu 112
Combining relations (2630) we get Garding's inequality. The use of Fourier series (or Fourier transforms) seems indispensable for the proof of Theorem 1. But in the classical case m = 2 integration by parts suffices. Indeed, assume that
LL1 L1
= 5 alk(x)
a2
a.r,ax; '
,
L2 a
L2 = Y a,(x) ax,  a(x)
HILBERT SPACE APPROACH, I
with E aik(x)
173
c E 6rj, c > 0. Then
au au
= SJ a,k ax. axk dx +
(u)Liu)o
a
2
>cI
If ax
au ku
t
dx
x
con st. Ilullo Ilulll
11TIUxillo
pos. const.
1 1 U1 1i
 const.
11U 11
o
and since I (u,L2u) I is easily seen not to exceed e const. IIu II
(1/e) const. IIu Ilo for every e > 0, Garding's inequality follows.

At the moment we shall prove Theorem 2 only in a weakened form. More precisely, we shall show that (22) holds for A > A0(t), A0(t) depending on t and L.
Let s be a fixed nonnegative integer. The differential operators K"L and LK" of order m + 2s are elliptic; hence, by Theorem 1, there are positive constants c;, c., depending on s such that Proof.
(u,K"Lu)o
:2! c'l Ilu ll8+,,,/2
(u)LKsu)o ? c1
IIu
112
 C2 Ilullo  c2' IIu Ilo
Using the first inequality together with (15, 6, 14, 4, 3) we have that flu II 8 I m/2
II Lu + Au II 8m/2 =
IIuII "+m/2
II K3Lu + AK"u ll sm/2
> (u, K"Lu + ).K"u)o > ci IIUII8+m/2  c2 Ilullo + A(u,Ksu)o
=
ci flu Iii+m/2
 cz IIu Iln + )'11U112 ? ci
IIu IIS+m/2
+ (2  c2) Ilullo ? ci Il u ll9 i m/2 for A > c2 and dividing by 11U 113. m/2 we obtain (22) for t = s + m/2. Similarly, IIuIIsm/2
IILu + Aull_sm/2 = IIK'ulls,,,/2 IILu + Aullsm/2
> (K"u, Lu + Au) o = (K"u, LK"K"u + Au) IIK_"u112
ci
= ci
11K"ull8+m/2  c2
IIUI12
8+m/2  c2
IIUII
+ A IIuI128
2" + . IIuII? ? ci Ilullo 8+m/2
for A > c? and dividing by IIu II s 1 m/2 we obtain (22) for t = s + m/2.
A complete proof of Theorem 2 will be given later.
L. BERS AND M. SCHECHTER
174
3.5.
Differentiability Theorem
Theorem 2 (in its weakened form) implies LEMMA 11.
For every t and for A > 0 sufficiently large the operator
L + A is a bounded linear onetoone mapping of H, onto the whole of H,,n and the inverse mapping (L + 2) 1 of H,_m onto Ht is also bounded, independently of A.
Proof.
By Lemma 9 the linear mapping L + A: Ht * H,,. is
bounded. If 2 is so large that (22) holds (at the moment this choice may depend on t), then Lu + Au = 0 implies u = 0, so that the mapping is onetoone. The inverse mapping is defined on
the range R, = (L + A) (H,) and is bounded by virtue of (22), the bound being independent of A for A > A0(t). Next, R, is closed. Indeed, assume that vi E R, and II v,  v 0. Then v! = Lu, + Au,, u, e H. By (22) we have that II u,  uk II t S c3(t) II Vi  Vk II tm , 0 for j, k + oo. Hence there is a uE Ht such
that Ilu,  u II, > 0. By Lemma 9, IILu1 + Au,  (Lu + Au) II tm
0. Hence Lu + Au = v and v e R. In order to prove that R, = H,_,n we use the adjoint operator L* (cf. Chapter 1). We note that (31)
(u,Lv)o
 (L*u,v)o = 0
for periodic C,, functions (proof by integration by parts) and hence
(by an obvious limiting argument) also for u c Hs, v e Ht with
s+t>m.
Assume that A is so large that (22) holds also for L* and that R, 0 H,_r. Then there is a w e H,_,n such that w 0 0, (w, Lit + Au),,. = 0 for all u in Ht (projection theorem, cf. Appendix I to this chapter). Hence 0 = (Ktmw, Lu + Au)o = (L*Ktmw + AKtmw, u)o for all u in H,, and in particular for all C,,, functions u. This implies
(for instance by (31)) that L*Ktmw + AKtmw = 0. Now, Ktmw e Hm_, and if A is large enough, (22) applied to L* shows
that K'mw = 0. Then w = 0, contrary to our assumption. THEOREM 3. If u is a periodic distribution and Lu a H u EHs+m
then
HILBERT SPACE APPROACH, I
175
This is a differentiability theorem. It implies that every (periodic) solution of Lu =f is a function if f E H_,,,, has L2 derivatives up to the order k if f e Hk_,,,, is of class C, if f e Hs with s >_ [n/2] + r + 1  m (Lemma 3), and is of class C,,,, iff is. In the next chapter we shall extend this result to nonperiodic solutions.
Set Lu ==f. By Lemma 7 the distribution u belongs to Hk for some k. Hence f + Au E Hm1n (k,8) and if A is large enough u = (L + A)' (f + Au) belongs to Hmin (k+m,s+m) Proof of Theorem 3.
Repeating the argument we see that u c Hmin (k+im,s+m) for j = 1, 2,.... Thus u c H,,+).,,. 3.6.
Solution of the Equation Lu = f
We shall show now that the equations (32)
Lu =f
(33)
L*v =f
form a Fredholm pair.° Here f is a given periodic distribution;
we already know that for f EH, every (periodic) solution belongs to He. In particular, all solutions of the homogeneous equations
Lu = 0 L*v = 0
(34)
(35)
are Cr,, functions.
Let A0 > 0 be so large that the bounded linear mappings M = (L + AO) 1: Ho H,,, and M* = (L* + An)1: Ho * H. exist (Lemma 11). Since H,,, c Ho we may, and shall, consider
M and M* as mappings of Ho into itself. By Lemma 10 these mappings are completely continuous. The homogeneous equations (34, 35) may be written in the form
uk0Mu=0, 6 See Section 1.5.
uA0M*u=0
L. BERS AND M. SCHECHTER
176
Furthermore, the operators M and M* are conjugate in Ho. Indeed, by (31), we have that (Mu,v)o = (Mu, L*M*v + AOM*v)o
= (LMu + ).0Mu, M*v)o = (u,M*v)o
Hence the FredholmRieszSchauder theory is applicable (cf. Appendix II to this chapter). We obtain THEOREM 4.
The homogeneous equations (34, 35) have the same
finite number of linearly independent solutions. THEOREM 5.
The equation Lu + Au = 0 has nontrivial solutions
only for a denumerable set of values ) with no finite accumulation point.
Indeed, the equation considered may be written in the form
uA'Mu=0,with 2'=io2. THEOREM 6.
Equation (32) is solvable if and only if (f,v) o = 0
for every solution v of (35).
Assume first that f e H, for some s ,,> 0. Then equation (32) is equivalent to the following equation in Ho: Proof.
u  AOMu = Mf By the general theory, it is solvable if and only if (Mf,v)o = 0 whenever v  AoM*v = 0. But this is the same as saying that (f,v)o = A0(f,M*v)o = 2o(Mf,v)o = 0 whenever v  ,10M*v = 0. Since v  A0M*v = 0 is equivalent to L*v = 0, the result follows in this case.
Assume now that f e H , for s > 0. Let t be an integer such
that m + 21 > s. Since LK' and K'L* are adjoint elliptic operators of order m + 2t, there is a Al such that LK' + ,11 and K'L + Al are onetoone linear mappings of Hm±l,_, onto H ,. It is easily Set M' _ (LK' + 21) 1, M'* = (K'L* + shown, as in the discussion preceding Theorem 4, that M` and M'* are conjugate in Ho. We now note that (32) is solvable if and only if (36)
LK'w = f
has a solution. Now (36) is equivalent to (37)
w  a1M'w = M !f
HILBERT SPACE APPROACH, I
177
and My E Hm+2ts c Ho. Hence we may again apply the abstract theory to conclude that (37) has a solution if (M f v)o = 0 for all v E Ho such that v  AIM(*v = 0. This is the same as saying that (32) has a solution if (f,v)o = A1(f,Mt*v)o = A1(M f,v) = 0 for all solutions v of (35). Thus the theorem is proved in this case as well.
Combining Theorems 4 and 6 (or reasoning directly from the general theory) we get Equation (32) is solvable for every f c H ,, if and only if (34) has only the trivial solution u = 0. THEOREM 7.
We can now complete the proof of Theorem 2. We first note that from Garding's inequality (21) it follows that for some A > 0, Lu + Au = 0 implies u = 0 whenever A > A [e.g., take A = c2 in (21) ].' Now assume that Theorem 2 is false. Then there exists, for some fixed t, a sequence {u5} in Ht and a sequence of numbers A; > A such that II u; IIt = 1, IILu, + A,uf llt_m , 0. Let Ao be so
large that the bounded mapping (L + A)1: Ht_.,,, , Ht exists for 2 >_ Ao (Lemma 11). Clearly A, < A. and we may assume, A >_ A. By Lemma 10 selecting a subsequence if need be, that A, we may also assume that {u,} is a Cauchy sequence in Ht_,,,. Now
ui = (L + Ao)1(Lu,  A,u,) + (A0  2,)(L + Ao)lu1 so that 0 for j, k  oo. Hence there is a u in Ht with II us  Uk Il t 0 and we have : Il u ll = 1, Lu + Au = 0 which is 11u1  u II t impossible.
The same method shows that whenever A is not an eigenvalue
(i.e., whenever Lu + Au = 0 implies that u = 0) (L + A) ': Ht is a bounded operator, the bound being uniform Ht_m on every closed set of A's not containing eigenvalues. Appendix I. The Projection Theorem
We shall consider some simple theorems in a real Hilbert space H. (H consists of elements u,v, ... , for which the operations of addition and multiplication by real numbers are defined and these 7 Recall that Lu + ..u = 0 implies, by Theorem 3, that u is a C,, function. Hence the GArding inequality can be applied.
L. BERS AND M. SCHECHTER
178
operations obey the usual rules of a vector space. Moreover, for every pair of elements u,v there is defined a real number (u,v) called the scalar product in such a way that (u,v) = (v,u), (u + v, w) =
(u,w) + (v,w), (Au,v) = A(u,v), (u,u) > 0 for u ; 0 for all u, v, w c H and real number A. In such a case we can define a norm 11u II = 1/(u,u), and it is easily shown that (u)v)I S IIuII IIuII 11u + v11 < IIuII + IIuII (1)
IIu+vII2+ Ilu v112=2(IIulI2+ IIvII2)
for all u, v c H. These statements are known as the Schwarz inequality, triangle inequality, and parallelogram law, respec
c H which tively. Lastly, H is complete: for any sequence satisfies ll u  um 11 > 0 as m, n  co, there is a u E H such that 11 u,, u11 *0asn oo.) A subset S of H is called a subspace if Au + ,uv is in S for all real numbers A, a whenever u and v are in S. It is closed if ueHand Ifu, u11 >0asn imply that u E S. First we prove
c S,
Let M be a closed linear subspace of H. Then for every u c H not in M there is a v e M such that LEMMA 1.
IIu  v 11 = g.1.b. 11u  w II WEM
Let d = g.l.b. flu  w 1j, w c M. Then there is a oo. From (1) sequence c M such that IIu  w, II , d as n Proof.
we see that 4 Ilu  (wm + wn) III + IIWm  wnII2
=
2( IIu
 w. 112 + Ilu wn 112) ' 4d2
as m,n > oo. Since(w,. + w,,) c W, 4 IIu

(wm + wn) II2 > 4d2
 wn II ' 0 as m, n > or,. Since H is complete, there is a v c M such that 11 wf,  v 11  0. This means that and hence
11 wm
IIu  v 11 = lim flu  wn 11 = d.
HILBERT SPACE APPROACH, I
179
(Projection Theorem). Let M be a closed linear subspace of H. Then for every u e H, u = v + w, where v e M and (w,M) = 0 [i.e., (w,h) = O for all h e M]. THEOREM 1
Proof.
If u c M, set v = u and w = 0. If u is not in Al, then
by Lemma 1 there is a v e Msuch that Ilu  v II = d, the "distance" from u to M. Now if f is any element of M
IIuvll222(uv,f)+22Ilfii2
IIu  L'IIZC llu  U  ?/ II2= for all real 2, in particular, for
2=(uv,f) Ilf112
Thus
Ilu  vl12 < Ilu  vll2 
2(u
IIf112
f)2 +
(u
f
II,f)2
and hence (u  v, f )2 < 0, which means that (u  v, f) = 0. Since f was any element of M, w = u  v meets the requirements of the conclusion of the theorem. COROLLARY 1.
If M 0 H, there is a nonzero element w of H such
that (w,M) = 0. Proof.
Take u not in M. Then by Theorem 1, u = v + w,
where v e M and (w,M) = 0. Clearly w
0.
A bounded linear functional Fu on H is a real valued function on H
which satisfies the following conditions: F(ui + u2) = Ful + Fu2; F(2ul) = 2Fui, 2 a real number; IF(u) I < K IIu II for all u e H and some K > 0. The smallest K for which the last statement holds is denoted by IIFII
THEOREM 2 (Representation Theorem; Frechet, Riesz). For every bounded linear functional Fu on H there is a uniquely determined element f e H such that Fu = (u, f) for all u e H and IIFII = IIf1I Proof.
Let N be the set of all v e H such that Fv = 0. N is a
linear subspace of H. For if vl and v2 are in N and Al and 22 are any real numbers, F(2ivl + 22v2) = ).1Fvi + 22Fy2 = 0. Moreover, Nis closed in H. For if {v,,} is a sequence in N and II v,,  v II 0, theniFvl and V EN.
180
L. BERS AND M. SCHECHTER
Now if N = H, the theorem is easily proved by setting f = 0. Otherwise there is a w 0: 0 in H such that (w,N) = 0 (Corollary 1). Therefore Fw  0 and for any u E H
FIu
Fwwl =Fu FwFw =0
and hence u  Fw w is in N. This means that
Iu Ftww,wJ
=
0
i.e., that (u, ') =Iw 11w112
Therefore
Fu=
(u
w Fw)
' IIwII2
1
and (w/III' 112)Fw is the element f in the conclusion of the theorem. If f' also represents F, then Il f _f'112 = (f  f ', f fl) _
(f f',f)  (f  f',.f ') = F(f f')  F(f f') = 0. Next, Ilf III = (,f f) = F(f) < IIFII
Il,f 11,
so that
11f 11
<_ IIFII.
For every e > 0 therc is a g E H with IIg Il = 1 andF(g) > IIFII Hence IIFII  E < (g,f) < Il f ll. Thus IIFII = 11f 11.
 E.
The following theorems will be found useful later. THEOREM 3 (LaxMilgram [1]). Let [u,v]
real valued function defined for pairs of elements in H which is linear in both u and v. Suppose it satisfies (2)
I[u,v]I < lull
(3)
Ilu 112
be a
IIvII
< K[u,u]
for all u, v in H. Then for every bounded linear functional Fu on H there is an f c H such that
Fu = [u,f] For fixed v e H, we have, by (2), 1 [u,v] I < const. II u II Hence [u,v] is a bounded linear functional in H. Thus (by Theorem 2) there is an element Sv in H such that Proof.
(4)
[u,v] = (u,Sv)
HILBERT SPACE APPROACH, I
181
Obviously, Sv is a linear mapping of H into itself. We observe that by (2) IISuII 2 = (Su,Su) = [Su,u] S IISuII
.
Hull
so that 11sull
< Hull
Furthermore, by (3), Ilu 112
<_ K[u,u] = K(u,Su) s K Ilu ll
IISuII
giving 1lull
This shows that S maps H in a onetoone way onto a closed subspace of itself. Moreover, this subspace is H itself. Indeed, if this were not so, there would be an element w in H which would
be orthogonal to every element of the form Sv (Corollary 1). This would imply that (w,Sw) = 0 so that [w,w] = 0 by (4). But this implies that w = 0 (by (3)). Now let Fu be any bounded linear functional on H. By Theorem 2 there is an f c H such that
Fu = (u,f) By what we have just proved, f = Sw for some w e H. Hence
Fu = (u,Sw) = [u,w] for all u e H and the proof is complete. THEOREM 4 (BanachSaks). Let be a sequence of elements in H such that II v II < K. Then one can find a subsequence lvn,} and an element v such that
vn1 + .
.
. + Unr
 v
31 0
v
asv oo. Proof.
We shall show that there is a subsequence {vn,} such
that (vn,  vnk, g)  0 as j, k  o for each g e H. It suffices to do this for g in the closure S of the linear subspace S spanned by the vn. For every g E H can be written in the form g = g1 + g2
L. BERS AND M. SCHECHTER
182
where g1 E 9 and (g2,S) = 0 (projection theorem). Thus (vn,,g) = (vn,,gl) + (v8,,g2) = (vn,,g1) and (vnr,g1) converges.
Now for fixed m, the real numbers (Un,vm) are bounded, and
hence there is a subsequence which converges. By the usual diagonalization process, we can find a subsequence {vn, } such that (Vn,) converges for each fixed m. Thus (v,1, f) converges if f is
in S. That it also converges for f in 9 follows from the inequality (vn,  vfk,f) I
<
I (Un),f  f1) I + I (vn,  Unk7fl)
+ I(vnk,fl f)I < 2K Of fill + I (vn,  UnL,fl)I
where f1 e S. For any E > 0 we can take f1 so close to f that 2K II f  f1 II < e/2 and then take j and k so large that (vn,  Unk, fl) I < E/2.
Thus (vn,  Unk, g)
0
for each g E H.
If we now set Fg = lim (g,vn), we see that F is a bounded linear functional on H. Thus there is an element v e H such that (vn!  v, g) > 0 for each g e H (representation theorem). Now set u f = v.n,  v. Then II u, II S K' for some constant K' and (u,,g) + 0 for each g E H. It is easily seen that we can now pick a subsequence {u, } of {u,} such that
I c t, ... , I (uj,, u,,+1) I c t
(uJ,'
Then
u;1+...+ur1 t
2
< [tK'2 +
3.1 and the proof is complete.
2(1
1,1
+2.2 1
e
K12
+2  U
HILBERT SPACE APPROACH, 1
Appendix II.
183
The FredholmRieszSchauder Theory
We shall now prove some wellknown theorems for completely continuous operators. It will be sufficient for our applications to
restrict ourselves to a real Hilbert space. However, all of the
theorems hold in any Banach space (with the appropriate definition of the adjoint of an operator).
As before we let H denote a real Hilbert space with scalar product ( , ) and norm II . We shall make use of the following elementary properties which are easily proved. II
LEMMA 1.
Every finite dimensional subspace of H is closed.
LEMMA 2.
A subspace Hl of H is finite dimensional if and only if
every bounded sequence in Hi contains a convergent subsequence.
We now consider a linear operator T which maps H into itself. Thus
T (au + ,8v) = xTu + pTv
for u, u e H and scalars a, j9. We assume that T is completely continuous. This means that whenever {u,} is a bounded sequence in H, the sequence {Tu,} has a convergent subsequence. We set L = I T, where I is the identity operator, and let N be the set of all u E H which satisfy Lu = 0.

LEMMA 3.
N is a finite dimensional subspace of H.
That N is a subspace of H follows from the linearity of L. To show that it is finite dimensional, let {u,} is a bounded sequence in N. Then Tu, = u,. Since T is completely continuous, {Tu;} has a convergent subsequence, and therefore the same is Proof.
true for {u,}. Hence N is finite dimensional by Lemma 2. Let M denote the set of those u e H which satisfy (u,N) = 0 [i.e., (u,v) = 0 for all v e N]. Clearly, Af is a closed subspace of H. LEMMA 4.
for all u e H.
There is a constant co > 0 such that lITull <_ co Ilull
L. BERS AND M. SCHECHTER
184
Proof. If this were not so, there would be a sequence {ui} in H such that 11ui II = 1 and 11Tu, II 1 oo. This is impossible since {Tui}
has a convergent subsequence. LEMMA 5.
There is a constant cl > 0 such that
cl' Ilull <_ IILull < cl (lull
for all uEM. Proof. The second inequality follows immediately from Lemma 4. If the first were not true, there would be a sequence {u, j in M such that 11u, II = 1 and Il Lug ll > 0. Since {Tui } has a
convergent subsequence, so does us = Tu, + Lui. Denoting this subsequence again by {u;}, we see that there is a u e M such that
u; , U. By Lemma 4, Tu, > Tu and hence u = Tu. But this means that u e N. Since u is also in M, u = 0. But lull =
= 1, giving a contradiction. For any fixed v e H, Lemma 4 shows that
lira llu1II
11v 1111 Tu 11
<_ const.
11u II
I (v,Tu) I
for all u c H. Hence (v,Tu) is a bounded
linear functional on H. By the FrechetRiesz representation theorem (Theorem 2 of the preceding appendix) there is a g e H such that (v,Tu) = (g,u). We set g  T*v. Clearly T* is a linear transformation of H into itself and satisfies (v,Tu) = (T*v,u)
(1)
Thus T** = T. LEMMA 6.
Proof.
T* is completely continuous.
Let {ui} be a sequence satisfying Ilui 11 < M. Then IIT*ui 112 = (T*ui,T*ui) = (ui,TT*u,) Ilui II IITT*ui II <_ coM IIT*ui II
giving IIT*ui 11 < c0M. Hence {TT*u,} has a convergent sub
sequence. Denoting this subsequence again by {TT*u,}, we have
IIT*(ui  ut)112 = (T*(ui  u,), T*(ui  u,)) = (us  u,, TT*(u,  u,)) < 2M IITT*(ui  u5) II 0 as i, j > oo. Hence T* is completely continuous. THEOREM 1. The equation v  T*v =f has a solution if and only iffEM.
HILBERT SPACE APPROACH, I Proof.
If v
185
 T*v =f, then (f,w) = (v  T *v, w) = (v,Lw) = 0
if w c N. Conversely, suppose that f e M. By Lemma 5, 11 Lull and
are equivalent norms in M. For u e M, I (u, f) I c const. and hence (u, f) is a bounded linear functional in M. Since M is a closed subspace of H, it is itself a Hilbert space. Thus by the FrechetRiesz representation theorem, there is a w e M such that 11 u 11
11 u 11
(Lu,Lw) = (u, f )
(2)
for all u e M. Now we claim that (2) holds also for all u e H. This follows from the fact that N is finite dimensional and hence closed. Therefore by the projection theorem (Theorem 1 of the
preceding appendix) every u E H can be written in the form u = u' + u", where u' E M and u" E N. Thus (Lu,Lw) = (Lu',Lw) = (u', f) = (u, f) since (u", f) = 0. Set v = Lw. Then (Lu,v') = (u, f )
for all u E H. This means that (u, v
 T*v) =
(u, f )
for all u e H. Thus v  T*v =f and the proof is complete. Set L* = I  T* and let N* and M* be the spaces corresponding to N and Al, respectively. We then have COROLLARY 1.
The equation Lu ==f has a solution if and only if
f E M*. Proof.
This follows from the fact that T* is completely
continuous and that T** = T. Define the operators L", n = 1, 2,
by the recurrence relations Li = L, L"'1 = LL" and let N" be the set of u e H ,
. . .
satisfying L'u = 0. LEMMA 7.
L' is of the form L" = I  T where T" is a completely
continuous operator.
Proof. We have 1 0(
') (
= I  (nT  (n)T2
f
irr
... ± T")
L. BERS AND M. SCHECHTER
186
and the operator in parentheses is easily seen to be completely continuous. COROLLARY 2.
Each N is a finite dimensional subspace of H.
This follows immediately from Lemmas 3 and 7. N,, C Nn,1.
LEMMA 8.
Proof.
Obvious.
There is a positive integer k such that N,, 0 Nn,, for
LEMMA 9.
n
We first show that N,, = Ni+1 implies N,,12= N,,. Then 0 = L"+2u = L"+1Lu. Hence
Let u be any element in Lu E
N,,.
This means that L"Lu = 0 showing that
u c N"+1 = N". Now suppose that the lemma is false. Then N. for all n. By the projection theorem (Theorem 1 of the preceding appendix) there is a sequence {un} in H such that un E Nn+1,
Ilu,, II = 1
and (u,,,N,,) = 0. We shall arrive at a
contradiction by showing that {Tu"} contains no convergent subsequence. To see this we note that T(u"  urn) = u,,  (Lu,, + Turn) Now for n > m, L"(Lu,, + Turn) = L"+1u,, + TL"urn = 0 Hence Lu" + Turn E N,, and IIT(u"
 urn)112 = Ilun 112 + IILu" + Turn II2 z 1
Hence {Tu"} has no convergent subsequence and the lemma is proved.
Let R (resp. R*) denote the range of T (resp. T*), i.e., the set of all v for which there is a u c H satisfying Lu = v (resp. L *u = v). LEMMA 10.
Proof.
If R = H, then N = 0.
Assume that R = H and that there is a uo : 0 such
...
be solutions of the equations Lu1 = uo, that Luo = 0. Let u1i u2, (which we can solve since R = H). Then L"un = Lug = u1i N,, for all n, uo 0 while Ln+1u,, = Luo = 0. Thus Nn+1
...
contradicting Lemma 9.
HILBERT SPACE APPROACH, I THEOREM 2.
Proof.
187
R = H if and only if N = 0.
By Lemma 10, it is sufficient to prove that N = 0
implies R = H. If N = 0, M = H and hence R* = H by Theorem 1. Since L* is completely continuous (Lemma 6), we see from this that N* = 0 (Lemma 10). This means that M* = H and R = H by Corollary 1. THEOREM 3. N and N* are of the same finite dimension.
That they are finite dimensional follows from Lemmas 3 and 6. Assume that N is of dimension n and N* is of dimension v. We shall show that v > n leads to a contradiction. By the GramSchmidt orthogonalization process there are elements u1, u2, ... , u in N and v1i v2, ... , v, in N* such that Proof.
(vi,vi) = big (u:)u,) = 6J, where 8i, is the Kronecker delta. Now define the operators n
Wu =u Su
Su = Tu  I z=1
It is easily checked that S is completely continuous.8 Now if Wu = 0, then it
0 = (v,,Wu) = (v,,Lu) + 1 (u=,u) i=1
for each j = 1, 2, ... ,
= (L*v,,u) + (u,,u) = (u,,u) n. But from the definition of W, this
implies that Lu = 0, i.e., u e N. But if u e N and satisfies (u,,u) = 0
for all j, then u = 0. Hence Wu = 0 has only the solution u = 0. Applying Theorem 2, we see that we can solve the equation Wu0 = Vn+1. But n``
(Vn+1)Wu0) = (vn+1,Lu0) +
i=1
(ui,u0) (Vn+1,vi) = 0
contradicting the fact that II vn+1 II = 1. Hence v > n is not possible. Considering the operator V
S*v = T*v + I (v,,v)ui s=1
8 In fact, S  T maps any bounded sequence into a bounded sequence in N*, a finite dimensional subspace. We then apply Lemma 2.
L. BERS AND M. SCHECHTER
188
and applying the same reason as above, we see that n > v cannot be, and the theorem is proved. THEOREM 4.
The equation
u  ATu = 0 has nontrivial solutions only for a denumerable set of real A having no finite accumulation point. Proof.
sequence
We shall prove that there does not exist a bounded of distinct values such that each equation
uATu=0
(3)
has a nonvanishing solution. For assume there were such a sequence. Let u 0 be any solution of (3). Then for each n, u1, ... , u, are linearly independent. This follows by induction. If the first k  1 are linearly independent and k
Ic,u, = 0 J
1
then
kA.c
0='k c,Tu;=I A
u,
j=1 F,
1
Subtracting, we have 0 j=t
showing that all the c; = 0 for j 0 k. Since uk 0, C. = 0 also and the u1i . . . , u, are linearly independent. Let En be the subspace spanned by ul,
. . .
, u,,. Then by the projection theorem
there are elements vn E En such that 11vn II = 1 and (v,,,Ei_1) = 0.
Moreover, if w E E,,, then w  ).,,Tw is in En1, for
and »
nxu
w).,,Tw=Ix,u;AnI j=1
J=1 A;
n1 j
1
.2 x;u5
HILBERT SPACE APPROACH, I
Now if the sequence {A,,) is bounded, then {T a convergent subsequence. But for n > m
189
must have
Amvm) = v  (v  A,,Tv,, + AmTvm) and
Amvm)II2 = IIvn112 + IIv  A,,Tv + AmTv,,, II2 > 1
since both v  A Tv and Tum = L are in Ei_1. This shows that m
has no convergent subsequence. Hence there does not exist a bounded sequence of distinct 1.,, and the theorem is proved.
CHAPTER 4
Hilbert Space Approach. Dirichlet Problem 4.1.
Introduction
We saw in the preceding chapter that the study of periodic solutions of elliptic equations lends itself to a particularly elegant treatment. Some of this machinery can be converted to give weak (or even "semistrong") solutions of boundary value problems and to establish interior regularity. But when it comes to proving that such solutions are smooth up to the boundary, new methods must
be devised. Several authors have studied this problem in great detail (Nirenberg [1], Browder [2], AronszajnSmith (cf. Lions [2] ), Schechter [5], Agmon [2], Peetre [1], and others). In this chapter we return to nonperiodic operators
L = I a, (x) D9 IPI <m
defined in a bounded domain V. We assume that L is elliptic and (for the sake of simplicity) that the coefficients a,(x) are in C., in the closure ci of W. We shall study the interior differentiability of weak solutions, formulate the Dirichlet problem within
the framework of Hilbert space, prove the existence of weak solutions and finally take up the problem of regularity at the boundary. 4.2. Interior Regularity
One property which follows easily from the results for periodic equations is interior regularity. THEOREM 1.
Let T be a distribution solution of L T = f in W. If f
is locally square integrable, then T is a function having L2 derivatives of order m. If f has L2 derivatives up to order t, then T has L2 derivatives up to order m + t. 1f f is infinitely dierentiable, so is T. 190
HILBERT SPACE APPROACH, II
191
First assume that T = u, a locally square integrable function. Then u is a weak solution of Lu =f. Since differentiability is a local property, it is sufficient to prove that in each subdomain 91 such that 91 c 9, u has L2 derivatives up to Proof.
order m + t, whenever f has L2 derivatives up to order t, If 91 is such a domain, let cc2 and 93 be such that cc1 92 C 93, cc3 C9. Let be any test function in g3 with in g2. Since u is a weak solution of Lu = f, we have f9, fo dx
t >_ 0.
q2,
=1
uLi O] dx
dx
for all test functions 0 in 9, where Li is a partial differential operator of order < m with C,,, coefficients having support in cJ3. Since all the integrands have support in c3, (1)
fiifw dx
=12
uLi w] dx
for all C., functions w which are periodic in some cube 2 containing T. We can alter u, f and the coefficients of L outside T. so that they become periodic in 2 in such a way that u and f remain in L2 while L remains elliptic having C., coefficients. and the coefficients of Li may be extended without Next, alteration. Denote the periodic extensions by u, f, L, etc. We have A
by (1)
A
A
A
f'=Lv+L,u where v = tic and L, is the formal adjoint of Li . Now by the theory of the preceding chapter f  L1u is in H,_,,,. Hence, by Theorem 3 of Chapter 3, v e H1. This means that u has L2 derivatives of the first order in cc2 '*,'Interpolating a domain between Y1 and cc2 and repeating the process in we see that , f  L,u is in H2_,,, and hence u has L2 derivatives of the second
order in a domain containing .11. This may be continued until L2 derivatives of order m + t are reached.
Assume next that T is a distribution. We shall show later (Corollary 2) that there is a positive integer k and a square integrable function u such that (2)
T[4]
=5
uAk4
dx
L. BERS AND M. SCHECHTER
192
for all test functions 0 in 4j. Assuming this for the present, we have
Jfc6 dr = LT[t¢] = T[L*¢]J uYL*0 dt
for all such j. Since the operator LA's _ (AkL*) * is elliptic and of order m  2k, we have by that part of the theorem just proved that u has L2 derivatives up to order m + 2k ; t. Hence by (2)
T[0] =
J
z
uq dx
for all test functions 0. Hence T = AAu and has L2 derivatives of order up to m ± t. The last statement of the theorem follows from Sobolev's lemma (Lemma 3 below). 4.3.
The Spaces H' and H'
We shall now describe some Hilbert spaces which will take the
places of the spaces H, of Chapter 3. For integers t > 0 and for C. , functions in ( we shall employ the scalar products (U) V) t = I J
dx
API
and the corresponding norms 11ul1t
We also set
=
(u,L)
=
(u,v)0
Completing with respect to these norms we obtain Hilbert spaces H° D Hl H2 As in the case of the H, spaces, functions in H' have L. derivatives in P up to order t. For each t, let Ho' be the closure in H' of the set of test functions
....
in f. Thus H is a linear subspace of H', and u e Ho if there exists a sequence {0J of test functions in qi such that 11 0,  uli e '
0. Clearly Ho' = H°. On the other hand, Hot is a proper subspace
of H' for t > 0. If a function u belongs to H;, one says that it vanishes at the boundary together with its derivatives up to order
HILBERT SPACE APPROACH. II
193
t  I in the L2 sense. The terminology is justified by the fact that if u has continuous derivatives up to order t  1 in and belongs to Ho, then these derivatives actually vanish on the boundary 'P, provided that 9 is sufficiently smooth. To see this, let {0Y} be a sequence of test functions such that III,  ull t  0. If w is in C1
in'° ff
[wD5q + &,,D,w] dx = 0,
.1=1,
n
Hence in the limit, j[wDiu
j = 1, ... , n
uD,u] dx = 0,
But if 0 is sufficiently smooth, this is equivalent to (3)
uw cos (n,x,) dS = 0,
j = 1, ... , n
fo where cos (n,x,) is the direction cosine of the normal to I and dS is the element of area. Since (3) holds for all w, u  0 in i. The argument holds for all derivatives of u up to order t  1. Some Lemmas in Ho'
4.4.
In many ways functions in Ho 'behave like the periodic functions
in H,. This can be seen from LEMMA 1.
Let 2 be a cube containing '9 in its interior. For u E Hnt,
set u in q A
U
0 i n 2 Then u, considered as periodic in 2, is in Ht and' !IuIIT = Ilulli,
be a sequence of test functions such that II0r  u li s  0. Extend each ¢,, to be The proof of Lemma 1 is obvious. Let
When necessary, we indicate the domain over which a norm is taken by a superscript.
L. BERS AND M. SCHECHTER
194
identically zero outside 9 and consider it periodic in 2. Then
III"  uli = II  ulle
0.
By employing Lemma 1, we can immediately carry over some of the results of Chapter 3. For instance, by (9) and Lemmas 3 and 10 of that chapter, we have
If 0 S t2 < s < tl and e > 0, then
LEMMA 2.
Ilull
<e
Ilullel
+
6`sts)J(tis)
IIu112
for all u c Hot. LEMMA 3.
Ck in
If u c Ha and t > [n/2] + k + 1, then u is of class
and max I Dpu l < cont. II u Il t Ipl
LEMMA 4.
For 0 < s < t every bounded set in Ho is conditionally
compact in H.
For functions in Ho we shall find it convenient to introduce the following scalar products and norms:
(u,v): _ 2 (t)JDPDPd v Ipi =t p
lute = (u,v)'12
where (PI I is the multinomial coefficient t !/p1 !
... p,,!
.
Integration
by parts gives
(u,v)e = (u,Aty) = (Atu,v) when u and v are test functions. LEMMA 5.
If 0 < s < t there is a constant c,e such that 1u18
for all u e H. Proof.
< c8e Iutt
It suffices to prove that II
for test functions W. Then
II o < const. 101,
in W. Extend 0 to be identically zero outside
41.
dx(x) =
axi 
HILBERT SPACE APPROACH, II
195
Applying Schwarz's inequality and integrating over T, we have IIdII 0
ao (1
< const *
11
axi 11o
which implies the desired inequality. COROLLARY 1.
In Ho' the norms Ilullt and Iult are equivalent.
This means that the expression Iult
Ilulla
lult
is bounded for all u LEMMA 6.
+
Ilullt
0 in Ho.
Let T be a distribution. Then there is a nonnegative
integer t such that (4)
I T[*]
cont. II
II
t
for all test functions ¢ in 9. Proof.
Following the reasoning of the proof of Lemma 7 of
Chapter 3, we note that if the assertion were false, there would be a sequence {0v} of test functions such that I T[O,] I > v 11 0, II Set Vv = 0v/v II 0Y II Then for every fixed s, II ip, II s = I10, II 8/v 11 0, II v  0
as v > cc. By Lemma 3, this implies that Dip  0 uniformly in Tr for each p. Hence T [v,v]  0. But I T [V,] 1, > 1, a contradiction. COROLLARY 2.
For every distribution T there is a nonnegative
integer t and a function u e Ho such that
f
T[¢] = guL1/ dx for all test functions 0 in 9. Proof.
By Lemma 6, there is a t ;.>_ 0 such that (4) holds.
Therefore T is a bounded linear functional on the Hilbert space H. Moreover, by Corollary 1, we may consider I It as the norm in Ho. Thus by the FrechetRiesz representation theorem (Theorem 2 of Appendix 1 to Chapter 3) there is a function u in Ho such that T[0] = (u,o)t
196
L. BERS AND M. SCHECHTER
for all test functions 0. Integration by parts gives
T[O] =J uL's6 dx the required representation. Note that Corollary 2 was employed in the proof of Theorem I. We end the section by proving a simple consequence of the BanachSaks theorem (Theorem 4 of Appendix I to Chapter 3). LEMMA 7. If ' v is a function in H°, and there is a sequence vn of functions in H' such that
II U. V 110  0,
!IUnii, <_ C
then v is in H' and II v II, < C.
Since H' is a Hilbert space, by the BanachSaks theorem there is a subsequence {'v,} of whose arithmetic Proof.
means converge to some element u e H'. But the arithmetic means
converge to v in H. Hence v= u c H'. 4.5.
The Generalized Dirichlet Problem
We are now in a position to formulate the Dirichlet problem within the framework of Hilbert space, an approach going back to Friedrichs. If u is a classical solution of the Dirichlet problem with homogeneous boundary values in a sufficiently smooth domain 5' (cf. Section 1.5), then u is in H,,"12 and satisfies (5)
(u,L*O) = (fo)
for all test functions in ', where L* is the formal adjoint of L.10 Now if V' and 0 are test functions, (6)
(LW,O) = (tV,L* ,) = [ju',o]
where (7)
NA =',, :_,»is I fsh,,.,,.(x)DP'pDr
dx
IP1I _
In That u e Ho %s is not obvious, but it is easily proved. Compare footnote 11.
HILBERT SPACE APPROACH, II
197
is a bilinear form involving derivatives of order < m/2 in either t, or 0. Thus (8)
< const. II1P II m/2 I
II
on/2
By completion it follows that [y,,o] is defined for W and 4 in H,)"12 and (8) holds for such functions. Thus our solution of the Dirichlet problem satisfies (9)
for all test functions 0. We now take (9) as the basis of our definition of
The Generalized Dirichlet Problem. Given f c H°; to find a u e H"/'' such that (9) holds. We know that a classical solution of the Dirichlet problem is a
solution of the generalized Dirichlet problem. Conversely, a solution u of the generalized problem which has continuous derivatives in SO of order m is a classical solution of Lu =f; this
follows from (5) by integration by parts. Moreover, if it has continuous derivatives up to order m/2 in jp and P is sufficiently smooth, we see that all derivatives of order <(m/2)  1 vanish on Thus u is a classical solution of the Dirichlet problem. Thus we may solve the Dirichlet problem by first solving the generalized Dirichlet problem and then showing that the solution has the required smoothness properties. While the problem of differentiability on the boundary is in itself very interesting, one should nevertheless remember that the generalized way of formulating boundary conditions is in many
ways natural. Thus it is well known that, even for Laplace's equation, the Dirichlet problem may not have a classical solution, that is, a solution which is continuous up to the boundary, for the
most general domain. The Hilbert space formulation of the Dirichlet problem, however, is applicable without any restrictions.
In a certain sense, this formulation of the boundary condition does the thinking which would otherwise have to be performed by the mathematician posing the problem. Consider, for instance,
the Dirichlet problem for the Laplace equation Au = 0 in 3 dimensions. We assume that the boundary 47 consists of smooth manifolds of dimensions 2, 1, and 0 (surfaces, curves, points).
L. BERS AND M. SCHECHTER
198
The proper way to prescribe boundary data is to give them only on manifolds of dimension 2 and to require the solution to be bounded on other boundary components. On the other hand, in
the Hilbert space approach, it seems that one prescribes the values of the unknown function on all parts of the boundary. But these values are prescribed not in the classical, but in the generalized sense, and it turns out that the solution obtained "forgets" the superfluous conditions and satisfies exactly the conditions indicated before. 4.6. Existence of Weak Solutions
In this section we shall solve the generalized Dirichlet problem.
Our main tool will be the Girding inequality which we state as THEOREM 2.
There exist positive constants cl and c2 depending only
on L and T such that (10)
(0, Lc6) >_ ci
121  C2 II¢II2
for all test functions 0 in I.F.
The proof of (10) is very similar to the corresponding periodic inequality. In fact an inspection of our proof of (21) of Chapter 3 will convince the reader that it can be carried over word for word to the present case. Employing (10) we can prove Let f be a function in H°. If .° > c2, there is a unique function u e Ho /2 such that THEOREM 3.
(11)
00) =(ff0)
for all test functions 0. Moreover, u has L2 derivatives up to order m. Proof.
Set
By Theorem 2, (12)
hPA, = [W,0] , 4(14) [u,u]' > ci
II U 11L/2
for all u e H0'7`/2. Moreover, by (8), (13)
[u,v]' < const. IIuII,02 IIvIlm/2
HILBERT SPACE APPROACH, II
199
Inequalities (12) and (13) allow us to apply the LaxMilgram lemma (Theorem 3 of Appendix I to Chapter 3). Since f E H°, I (f v) I < 11f 110
Il v II O
<_ const. Il v 11 m/2
and (f,v) is a bounded linear functional on
H012. Hence by the
lemma, there is a u c Ho'12 such that
[u,v]' = (f,v)
(14)
for all v c H,/2. By (6), (14) implies (11). The uniqueness follows from the fact that any other solution must satisfy (14) also. Hence, the difference w satisfies [w,v]' = 0 for all v e Ho /2. In particular,
this holds for v = w. Thus [w,w]' = 0 which implies w = 0 by (12).
Let us call the solution in Theorem 3 the generalized solution of the Dirichlet problem for Lu 4 ).Ou =f By (12) and (14) we have I1ull2nf/2
< const. If II0 Ilulj0
Setting u = (L + A0) 'f we see that II (L + )0) '.f 11 m/2 = const. If 110
Thus by Lemma 4, (L + A0) 1 is completely continuous. This allows us to make statements about generalized solutions of the Dirichlet problem for the equations
Lu =f L*v =f
(15)
(16)
Lu = 0 L*v = 0
(17) (18) THEOREM 4.
Equations (17) and (18) have the same finite number of
linearly independent solutions. THEOREM 5.
The equation Lu + Au = 0 has nontrivial solutions
only for a denumerable set of values A having no finite accumulation point. THEOREM 6.
Equation (15) has a solution if and only if (f,v) = 0
or every solution v of (18).
Equation (15) has a solution for every f E H° if and only if u = 0 is the only solution of (17). THEOREM 7.
200
L. BERS AND M. SCHECHTER
The proofs of Theorems 47 are identical to those of the corresponding theorems for periodic equations (see the end of Chapter 3). 4.7.
Regularity at the Boundary
We are going to show that under further regularity assumptions a generalized solution u of the Dirichlct problem is a on f and
smooth function in
satisfying the equation and boundary
conditions. Assume that is of class C,,,, and that f is in C,o in Let u be a solution of the generalized Dirichlet problem with homogeneous THEOREM 8.
boundary conditions for Lu  f. Then u is in C,,, in 1. Since U E Ho'g' it will follow from the discussion in Section 4.3 that u and all its derivatives up to order (m/2)  1 vanish on ik.
Hence u is a solution of the Dirichlct problem in the classical sense.
The proof of Theorem 8 is based on a special result referring to a domain S2, which is the intersection of the ball Ixi < r and the halfspace x > 0. We consider SD, as contained in a halfcube E, the intersection of x > 0 and a cube 2 with edges parallel to the coordinate axes. Let V be the set of C. (not necessarily periodic) functions in E which vanish identically near x,  0, and let V' be the set of C,,, functions which vanish identically near the remaining faces. In
addition to the norms of the spaces H' in Id' we shall find it convenient to employ the following "negative" norms: (19)
IIUII_8 =
i(v,w)
I
Ilwlls
which we shall consider as the counterparts of the periodic negative norms over a. We now have THEOREM 9. _Let L be an elliptic operator of order m = 2m' with C, coefficients in E. Let t be an}' integer > m' and set s = max (m, t
Ij v is a function in Ht n H."', f l Cs in S2, which vanishes near I xI = r and such that IILv II,_,,, , 1 is finite, then v e H'1 in 52,.
HILBERT SPACE APPROACH, 11
201
The proof of Theorem 9 will be given in the next section. Here we shall make use of it in the proof of Theorem 8. We already know that u has derivatives of all orders at every interior point of V (cf. Theorem 1 ). I t remains to show that u is in C,, up to the boundary. Since regularity is a local property, it suffices to prove that for any boundary point x0 there is a neighborhood .:1 such that u is in C, in n ,1,(x0). Since is of class C,,, we can find a neighborhood .14''(x0) so small that it can be mapped
in a onetoone C., way onto a ball Ixi < R in such a way that n A *(x0) maps into the hyperplane x,, = 0 with x0 going into the origin. Under such a mapping the ellipticity of L is preserved. Assume that the image of 1i l ;1 "'(x(,) is Q,,. Now by Sobolev's theorem (see the remark following the proof of Lemma 9) it suffices to prove that in some S2 r < R, u is in H' for all 1. We do
this by showing that for any r < R and e > 0, u c H' in SZ, implies u e H' I' in S2,_F.
This may be done as follows. Let be a C,, function in f2n which is identically one in SZ,_, and vanishes near Ixi = r. Since u c H' in S2 so is v = u. We wish to apply Theorem 9 to v (since uEH
we need only consider t > m'). In order to do so we
must verify that II Lv II tm+i is finite. But
Lv=Cf+L'u where L' is an operator of order <m. Hence IILvIItm,, < const. (Ill IItm+i +
is obtained by
I
integration by parts in the usual way. Since O u II t is assumed finite, the same is true for IILvII: ,,, i Hence Theorem 9 applies giving v E H'' in Q,.. Since is identically one in S2,_ this means that u E H'' in S2,_E and the proof is complete.
In following the proof, we see that if we had assumed only that f e H'k in 9, we would have been able to conclude that u c Hm, k. In fact we have proved the inequality Remark.
IIuIIn.jk
< const. (Il f IIr + IIuilo)
This is one of a spectrum of "coerciveness" inequalities introduced
by Aronszajn [2] and studied by several authors (Browder [2],
L. BERS AND M. SCHECHTER
202
Nirenberg [4], Schechter [6], Agmon [3], Guseva [1], Koselev [1], Slobodeckii [1], and others). 4.8.
Inequalities in a HalfCube
First consider a cube 2 with sides parallel to the coordinate axes. Employing the notation of Chapter 3, we state LEMMA 8.
(20)
Let m and t be any positive integers. Then
IIDnvlla < cont. (flD1v'2 n lf1n+1 +
n1 1JDwll t =1
for all periodic Cx functions v. Proof.
If v = E at eit'', then e . ()t Iati2
IIDnvII = const. IIDnvili m+1 =
cont. Jen m(1 +
II Dtv II i = const.
f (1 +
. C)tmr1 Ixt12 ?') t I xtl2
Hence (20) is equivalent to the inequality t ` n < cont. [ im ( 1 + '
)"
'+
n1 a=1
which is true for any positive m. (We have tacitly assumed that the sides of 2 equal 2ir; this involves no loss of generality.)
Next, consider the intersection E of 2. with the halfspace xn > 0. We recall the definitions of the classes V and V' given in the previous section. LEMMA 9.
If the norms are taken over ' E, then (20) holds for all
v E V'. Proof.
We employ a device due to Lions of extending
functions in V' to be in C. and have compact support in the whole of .2. They then can be considered periodic and inequality (20)
applies. We then show that the periodic norms in 2 of the extended functions can be estimated in terms of the corresponding norms over E.
HILBERT SPACE APPROACH, II
203
To carry out the procedure, if v e V', consider it (for the moment) to be zero outside Z in x > 0 and set X" > 0
v1(x',xn) = v(x',xn)
_
8}1
x,, < 0
.Z.v(x', k=1
... , xi_,) and s = max (m, t ± 1). The constants
where x' = (x,,
Ak are so chosen that v1 is in C. in 2, i.e., that RTL
(21)
k=1
j = 0, 1, ... , S
(k)'2,, = 1
Thus v1 has compact support in .2, and ID'vlz dx
ID11v1I2 dx <
(22)
f.2
const.JP
Considering v, periodic in 2 we see that (20) holds for it. Hence, if all of the norms in (20) are nonnegative (i.e., if t > m 1), it follows from (22) that (20) holds for v.

It remains to prove that for t < m  1
,, j <_ const. 11Dn'vlli nn1 (The norm on the left is the periodic norm for 2; the norm on the right is that of the type (19) for Z.) In order to do this we (23)
IID,,v, ll
first note that for v e V' and wEV (24)
I (v,w) I <
Il v ll tm.+ 1 II W II
m t_1
By completion, (24) holds for w e C,,, in E satisfying" (25)
onxn=0,
D;,w=0
j=0,1,...,mt1
Now let g be any periodic C. function in 2. Then (Dn v1,g) 0
= f D'; vlg dx
=
f D; vlg dx + f
Dnvlg dx
11 This may easily be seen by first observing that (24) holds for C,n_
functions which vanish near x,, = 0. Then if w satisfies (25),
W(X', x  e) for x > e, = 0 for 0 < x,, <. e, is such a function and
IIwt wll,,,_t1 0 as
e0.
L. BERS AND M. SCHECHTER
204
Converting the last integral, we obtain (Dn'v1,g) = JDV&' dx
where g'(X',Xn) =
k1
k
By (21) we see that w  g' satisfies (25). Hence by (24), C III
I
vlll m+l llg'll
< const. II D;,'v ll
"1+1 Il; li
l
1
Since this holds for all periodic g, it follows from Lemma 5 of Chapter 3 that (23) holds. This completes the proof. Remark. The extension of v employed in the proof of Lemma 9 shows that Lemmas 2 and 3 hold for functions in F. An immediate consequence of Lemma 9 is LEMMA 10.
Let L be a partial differential operator of order m with
coefficients continuous in Z. If the coefficient of D;;' in L does not vanish in Z, then
(26)
IIUII
_r
< const.
n1
r
IID,vll
I
IlUllu)
,=1
for all v E P. Proof.
We know that L = aDn` + L1 + L2, where a
0,
L1 involves derivatives of the form D" with IpI = m, pn < m, and L2 is of order <m. Set ao = min lal > 0. Then (27)
IID','vlll
1
ao
(IuLvll1,n,1
,
IIL1vlll_mY1 + IILZvll,m+1)
If the norms are positive we see that (28)
IIL1vll m+1 <
const."I 1 IID,vIl 11
In addition, by the remark above, for any E > 0 (29)
IlL2vll l
1 ` e
lip ;I
r. 1  const. llvllo
Inequality (26) now follows immediately from (20, 27, 28, and 29).
HILBERT SPACE APPROACH, II
205
If the norms are negative, let w be any function in V and consider the expression (L1v,w). We can integrate by parts, throwing derivatives up to order m  t  1 onto w in such a way that every derivative remaining on v involves some x i : n. Then
n1 I (L1v,w) I < const. Il w ll »,t1
II D,v II t i=1
Since this is true for all w e V, (28) holds. Also, in the expression
(L2v,w) we can integrate by parts, throwing derivatives up to order m  t  1 on w. The derivatives remaining on v are of order
Hence II L2v II
t, . 1
const. II v II t
and (29) follows from Lemma 2. The proof is thus complete.
We now observe that (26) will hold with the same constant in any subdomain consisting of the intersection of 2, with the half
space x > h > 0. Letting a o 0, we have Set s = max (m, t , 1) and suppose that v is in C, n HO in E and vanishes near all faces except x = 0. Assume that D,v E Ht in for i = 1, ... , n  1, and that IILvll,,,t+1 is finite. COROLLARY 3.
Then v E H`+1 in 2.' and (26) holds.
We use induction on I and show first that the assertion is true for t = m'. Let a' denote any difference quotient in the x, direction, i , n. Then Proof of Theorem 9.
L,,,r
L8;'v
where L,,, is the operator whose coefficients are the difference quotients of those of L. Thus IILb;'z'il
,n
IIa (Lz')'1<. IILz'11
 I'L,,,z ll_ const. IIi'11,,,
and is finite. (Integration by parts was used in obtaining these estimates.) Hence there is a constant C such that (La'v,g) I < C iigll
206
L. BERS AND M. SCHECHTER
for all g E V and, a fortiori, for all test functions in 52,. Since v E Ho"", so is b,v for h sufficiently small. Hence by (6) I[bhv,g]l s C Ilgllm'
(30)
for all test functions g in S2,. By completion (30) holds for all g E Ho". We may therefore take g = bhv. Now by Garding's inequality [cf. (12) ]
Hence
[b"v,bhv] ? cl ci Il
61
'V112'  C2 IIb,vII
bhvllm < C Ilbhvll,,' + C2 Ilb"vIIO
showing that there is a constant C' such that Ilbhvllm' < C'
Now 116S  D'v II
0 in K2r_E for any e > 0 (cf. the end of
Appendix II to Chapter 1). Hence, by Lemma 7, D,v E Hm' in S2,_ and IID,vllm' s C'
Letting E  0, we see that D,v E H7" in Q,,. Once this is known, we
can apply Corollary 3. We therefore conclude that v e Hm'+i in QT.
Now assume that the theorem is true for t  1 > m'. We have
LD,v = D,Lv + L'v where L' is an operator of order <m. Hence IILD,vII:m < IID,LVIlt,n + IIL'vllt
s IILvIItm+i + const. IIvIIi
This is obvious if t >_ m. For t < m one must integrate by parts _,,, is finite. Moreover, if i 0 n, D,v E Ho m' and since v e H', D,v E Ht'. Hence we may
in the usual fashion. In any event, II LD,v II
apply the induction hypothesis to D,v, obtaining D,v E H'. Another application of Corollary 3 gives v c Ht+' and the proof is complete.
HILBERT SPACE APPROACH, II
207
Analyticity of Solutions
Appendix.
Hilbert space methods can also be used to give a proof of the analyticity theorem mentioned in Chapter 1. Assume that f and the coefficients of L are real analytic in qi and that u is a solution of Lu =f (we may even assume u is a weak solution or a distribution). Then by Theorem 1 of this chapter, u is in C.0 in q. We must show that for each point x0 in 9 there
is a small ball Ix  x01 < b in which u may be expanded in a power series. By a translation, we may assume that x0 is the origin. We shall find it convenient to use the following norms and seminorms Il w Il a,r
=
w18,r
=
DDwl2 dx
1P1j xl
Let R > 0 be such that lxl (1)
I
Ir=a J xI
ID'wl2 dx
R is contained in 9. By hypothesis
1(IDsa91+ID8fI) <Mok!Bk
k=0,1,2,...
181=k
in IxI < R for some positive numbers M0, B. Let r < R be such
that R/2 < r < R, and h > 0 such that r + h < R. Let a(T) be a fixed Cc function of one variable which equals 1 for T < 0 and vanishes for T > 1. Set fi(x) = tr`IxI
 rl
h
Clearly = 1 in IxI < r and vanishes for IxI > r + h. It is easily checked that (2)
ID8l ( S cls1 h1q'
in IxI < R, where the constant c181 depends only on n, Ist and Q(T). If w is any C. function in IxI < R, set v = w. Then v is a test function in lxi < R. We may therefore consider it to be periodic in a cube containing IxI < R. We may alter the coefficients of L outside lxi S R so that they become periodic also. We now can apply Theorem 2 of Chapter 3 to obtain t = 1, 2, .. (3) Ilvlit,r+h < ct(IILvIItm,r+h + Ilvlltm,r+h),
L. BERS AND M. SCHECHTER
208
where we have made use of the fact that v vanishes for Ixl > r + h. Now by Sobolev's inequality (cf. Lemma 3 of this chapter) there is
an integer N (>m) such that max IvI < x(n,N,R)
(4)
Izi _x
Employing the fact that Ivl g,1; and IIVII:,R are equivalent norms
for test functions (Lemma 5 of this chapter), we see that (3) implies (5)
C Const. (ILvi \.m,r
IIVII.\',r+h
!h
+
Ivl.V.,,+h)
Now if Isl = N  m, there are C,, functions bxo such that
DBLv = D3Lw + Hence (6)
.1 p'to
bD,D.Dv'4DD"w
N
II wll v,rh <
C(ILwI.Vm.r+h
+
h'
Ilwll,;,r+h
j =1 where the constant C does not depend on w, r, or h. Now it is easily verified that IDBLD'ui
)
(I41)
ID"TgfI + Kl IpI
lit Is's"I
Iq'4qI =IqI
14
I)
q' 7t o
x ID$ +q apI I De
Fq'+puI
where K1 depends only on Isl. Hence by (1), lLD`'uI_\'m,rlh <
Mo(N + IQI

m) !
Bn+IqIm
Xm Iq
lyl){i +J)! B`' I 0 i=1 lq'I=IqIi
+ K2
IIDq'ull:.r+h
Inserting D"u for w in (6), we have (7)
1 II Dqull.\,r+h < Cl WO(N + k  m) !
B`+km
IqI =k
\r
hs IqI =k 1
I k
II D''u II N,,rth
1
(k) (r + J) ! B''
S=1 ,=0181 =ki J
where the constant Cl does not depend on u, k, r, or h.
I1Dqull.vi,r+h}
HILBERT SPACE APPROACH, II
209
We now claim that it follows from (7) that there are constants
M and k such that for each r, R/2 < r < R and all integers
j>0
II D`u II:V,rh, < :11j !
(8)
where h, = (R that
J=
R  r)
ICI =
 r)lj. Once (8)
0, 1, 2,
...
is proved, it follows from (4)
I max ID`uI < a(n,N,R)Mj! (R
A
...
j = 0, 1, 2,
r
ItI =j 1XI
which immediately implies analyticity of u. It therefore remains to prove (8).
We proceed by induction. Clearly, (8) can be made to be satisfied for] = 0. Thus let k be any integer > 1 and assume (8) is satisfied for all j such that 0 < j < k. We shall show that it holds for j = k. Let r be any number between R/2 and R and set
ho = R  r. Then h, = ho/j. Now by (7) and the induction hypothesis II
IQI =
k
<
Dv UII c,rhx
C11 Mo(N
+ k  m) ! B` +k 'k
.IV
+ j=1 I hk'M(k j) ! k J'n: `k l
lx
+j)!Bi,M(k i j)!
lk (N +k m)! M k! (kj)! o
Mk ((a ,
rn (k J. k j

/Z
Bx ''oh X11
 i j) ! (i +j)! (B(ho  hk) (k j)! j! m ('2Bh0'v
<Mk! (h) o k
+ C2
(h 0
/z
1
+
J
ho  hk,J
+
,>i
A'
A
hk)l'
2B(ho l
1
1
A
k D
(
ho
J
ho
(k
 j) ! k!
hZ\)k )
I
kk
(k 
1)k'
210
L. BF.RS AND M. SCHECHTER
where we have used the fact that
(Nm+k)! SC22kk! and (i +j)! SC22j!,
i SN m
Now there are also constants C3 and C4 such that
(k j)!
kk
k! k
(k1)k,SC3
1k
S C4
k
Hence
I IIDuIIv.T+n <Mk! I1.1' M' 2BNm M
.
(2B/zo)k
t
+
N C3
+
N
C
2B(ho2 hk)
4 7
i1
If we choose M and A so that (9)
(10)
M>
3MOC2BNm
2 z max (3NC3,6C2C4BR)
it follows that (8) holds for j = k. The proof given here is essentially that of MorreyNirenberg [1].
CHAPTER 5
Potential Theoretical Approach The Hilbert space approach described in the preceding two chapters is very elegant, but this elegance is paid for by neglecting certain finer aspects of the theory. Various questions in the theory of linear elliptic equations cannot be answered by the L2 theory, and this applies particularly to questions which are related to the more difficult theory of nonlinear equations. In order to investigate
these questions, one must use tools from potential theory. We devote the beginning of this chapter to a description of these tools. Fundamental Solutions. Parametrix
5.1.
The potential theory approach to elliptic equations is based on certain simple integrodifferential identities and on three sets of
inequalities for convolutions of functions with powers of the distance from a fixed point. We shall discuss the identities first.
Let O(x) be an infinitely differentiable function with compact support [x = (x1, . . . , x,), n >_ 2]. Then Fundamental Identity.
(1)
O(x) =
(la)
O(x)
AO(y) dy
(n > 2)
IX yln2 O(3') dy
= c.Af
f
I
y
(n > 2)
2
fi(x) = c2 &/(y) log Ix yl dy
(la') fi(x) = where Cn
_
c2+(y) log Ix yl dy
(n = 2)
(n = 2)
1
F(n/2)
2(n  2)7rn12  (n  2) Q,,
_ C2
1
27T
(Q is the surface area of the n dimensional unit sphere). 211
L. BERS AND M. SCHECHTER
212
The proof is well known. We consider only the case n > 3. By Green's formula every sufficiently smooth function O(x) can be represented as the superposition of three potentials c"
'fi(x)
y1)2 d>'
J`M;)
fo, IX
1
n2
an ix ac(y)
1
an
Ix  yl,l2
dS
dS
Here I is a domain with a sufficiently smooth boundary x is a point of 1, ajan represents differentiation in the direction of the outward normal, and dS is the area element on k For a function
of compact support, we may take 9 to be a sufficiently large sphere. In this case the boundary terms disappear and identity (1)
results. This identity can be rewritten, in terms of distribution theory, as
(n > 2) Oc2 log IXI = b(x) (n = 2) where b(x) is the symbolic Dirac function, i.e., a distribution defined by the relation Ac.,
Ix12n
= b(x)
J(x)6(x) dx = 0(0) for every test function O(x). Now (la) follows immediately from (1), since O(x  z) dz = f ao(x  z) o r 0(y) dy = ASJ dz x
IX
IZI12
yln2
=f AM ') IX
yj 11
2
dy
J
IZIn2
= c 'fi(x)
From (1) we obtain, integrating by parts, the identity (2)
O(x) = const.
y' a4 dy
f X` _ Ix J yI" ay,
which represents a function with compact support in terms of its derivatives. For an elliptic operator L of order >2 with constant coefficients
and only highest order terms, one can construct explicitly a
POTENTIAL THEORETICAL APPROACH
213
function J(x) related to it in the same way as (xlzn is related to the Laplacean A. More precisely, the function J (x) will have the following properties. (i) J (x) is a real analytic function for I x I 0. (ii) If n is odd or if n is even and n > m, then
(3)
J (X)
I(x)
where w(x) is positive homogeneous of degree 0, (w(lx) = w(x), t > 0). If n is even and n S m, then
J (x) = q(x) log
IxI +
I(x)
where q is a homogeneous polynomial of degree m  n. (iii) The function J (x) satisfies the equation
L0J (x) = b (x)
(4)
so that for every C., function with compact support (5)
f
f
0(x) = [Lo/(y)]J(x y) dy = Lo s6(y)J(x y) dy
The result is classical; we carry out the construction only for the case of an odd n > m and for m = 2 and all n. (As a matter of fact, we shall assume that n is odd and exceeds m whenever it is
convenient to do so during this chapter. This assumption, howdenote the characteristic form of ever, is not essential.) Let Lo, i.e., let a,,DP, I a,E" Lo = IPI =»6
IP1 =m
Consider the function
Ix = x1E1 + ... +
It is obvious that this function is positive homogeneous of degree m + 1 and real analytic for where x 
x 0 0. It is easy to see that Lol(x) is positive homogeneous of degree one and invariant with respect to any rotation about the origin. Hence LOI(x) = const. Ixt. Now set J(x) = cAn+112I(x), where c is a constant. This function has properties (i) and (ii).
L. BERS AND M. SCHECHTER
214
That it also has property (iii) for an appropriate choice of the constant c follows from the fact that L0J(x)
= Loci"+1/2I(x) = cA"+1I2LOI(x) = c0"1/2 const. IxI = const. A(An1I'2 IxI)
= const. A x12" = b(x)
When m = 2 we can write down the fundamental solution just as easily. In fact if
=.I a,,$,s, let A denote the cofactor of a1, in the determinant la,,l. Then
J(x) = c(,J
A.1x,x,)2,7/2
J (x) = c log (I A1,x,x,)
n>2 n=2
One readily verifies that these expressions satisfy the required stipulations.
The function J (x) just constructed is called a fundamental solution of the differential equation Loo = 0. More generally, for any homogeneous elliptic equation L¢ = 0, a function J (x,)r) depending on a parametery is called a fundamental solution if it satisfies the equation L.J (x,y) = b(x  y)
It should be remarked that bounds on the derivatives of w (x) in (3) depend only on n and the ellipticity constant of Lo, i.e., on
min IQ( ) IFI=1
Fundamental solutions for elliptic equations have been constructed by many authors, the most important results being due to John whose book [1] also contains an extensive bibliography. In
particular, fundamental solutions are known to exist for any equations with analytic coefficients (in which case the fundamental solution itself is analytic) and for any equation with C,o coefficients (in which case the fundamental solution is itself C.) as well as under much weaker assumptions. As a matter of fact, it
is probable that the fundamental solution can be constructed
POTENTIAL THEORETICAL APPROACH
215
for any equation with Holder continuous coefficients, though as far as we know, this has never been proved in the literature.
Fundamental solutions play an important part in the theory of elliptic equations. In particular, once a sufficiently nice fundamental solution has been constructed, one can easily prove analyticity and differentiability of solutions. An arbitrary solution
of an equation defined in a domain 9 with a sufficiently nice boundary ! can be expressed in terms of its Cauchy data at the boundary (values of the function and its derivatives up to the order m  1) in terms of the fundamental solution of the adjoint equation. If a fundamental solution satisfies appropriate boundary conditions (in which case it is called Green's function), a solution can be expressed in terms of fewer boundary data. Fundamental solutions also play a part in the integral equation approach to boundary value problems. This method is classical for the potential equation and for second order equations in general;
an excellent account will be found in the book by Miranda [1]. The integral equations method has recently been extended to
equations of higher order (Agmon [4] and others). But our concern here will be only with the use of the simplest fundamental
solutions, namely the functions J (x) constructed above. These
applications are based on a simple device due to Korn and Lichtenstein.
To an elliptic operator of order m with variable coefficients a,(x) and to every point x0 we associate the "tangential operator"
=
v
lpl=yn
which is a homogeneous operator with constant coefficients. We shall denote by J.,°(x) the fundamental solution of the equation Lx°4 = 0 constructed above. JZ° is called a parametrix for the equation Lc = 0 with singularity at x0. We shall denote by Sx0
the operator which takes a function O(x) into the function ti = Sx00, where (7) V (x) =fix°(x y)o(y) dy Finally, we define the operator Tx° by the equation (8)
Tx° = SX°(LZ°  L)
L. BERS AND M. SCHECHTER
216
Since S 0L, ,
= 1 (= identity) on functions with compact support
and similarly L 0ST0 = 1, by virtue of identity (5), we have the following If 0 has compact support, then
"LEMMA" A.
= Tx0 ± Sx0L¢
(9)
and if 0 = Txa¢ ± Sxu f
(10) then
is a solution of the equation Lc = f.
The quotation marks around the word lemma are in recognition
of the somewhat cavalier way in which the lemma has been stated. It goes without saying that in order to become a real lemma,
it must be supplemented by a statement on the assumptions involved. In applications to be considered, all this will be selfevident. The proof of A is clear. Equation (9) follows by noting that on
functions with compact support T,0 + S= L = 1. On the other hand, assuming that (10) holds we have (Tx0 ; Sx L)o = Tx0o Sx0 f hence, S_L4 = S.,, ,f and Lq = L.,,, S,.0L0 = L,,Syo f =f. The applications of Lemma A are based on certain inequalities which are best stated in terms of certain norms. Some Function Spaces
5.2.
We consider first C X functions defined on the closure of a boun
ded domain1i. For these functions we set
0,
(jIvdx)
1/r
(r > 1)
l.u.b. ICI
H..gr[o] = l.u.b. ,._0
Ix,  X "l
0 < x < 1
(diam V)' max IIDp0!Ic,(9), In!
k = 1, 2, .. .
POTENTIAL THEORETICAL APPROACH II
(diam V)k" max H,,,,[Dp0],
IICktx() = It'IICk(l)
1PI =k
k=o,1,..., II0IIJi ()
Il
217
=
(diam
llNk(W) = I (diam q)Iv i ,nlr IPlsk
0<x<1
max II DI X11 La(s), IpI k k = 1)
IIDD0IILr(F)'
2,...
k = 1, 2, .. .
All these norms satisfy the axioms of a normed space. By completing the space of C. functions in each of these norms, we obtain the Banach spaces
LrM, CkM, Ck+x(
),
BB(9),
Nk(9)
The space NC(I) consists of functions which have Lr derivatives up to order k. The space Ck(I) consists of functions which have derivatives up to order k. The space Ck+«( Y) consists of functions
which have continuous derivatives up to order k, the highest derivatives being uniformly Holder continuous with exponent a. The space Bx(Ii) consists of functions which have continuous derivatives up to the order k  1 and Lr derivatives of order k. If the domain 1 is the ball of radius R (i.e., IxI < R), we shall write Lr(R), Ck(R), etc., instead of Lr(IxI < R), Ck(IxI < R), etc. We list the following properties of the norms which we shall need later (1 la)
II0IIrr(s)
(11b)
II4IIc,(f) s
(l lc)
IIOIIBk(f) <_ II
I!.\ (y )
for9c 9'. (12)
II4IIL,M < II0IIL,(s) II
UC,(S) < II0IIC,(#)
(14) (15a)
II4(Rx) IIc,(i)
=
(15b)
II#(Rx) JIB"(,)
=
(15c)
II0(Rx)
for s < t
(const. depending on s) I'!1C() IIWllc,(c4) Ilc,(R)
II0(x) II0(x) Il.Vk(ft)
218
L. BERS AND M. SCHECHTER
The proofs of (11), (13), (14), and (15) are obvious; (12) is the classical Holder inequality. We remark that it would be easy to modify the definition of the norm II II c,cy> so as to replace (14) by
the nicer inequality IlOPllc,(Ci)
In this case C3(c') would be a Banach algebra. We now discuss certain properties of the norms which are true only when the domain in question is sufficiently regular. For our purposes the following conditions will suffice. We shall say that 9 has the Cs extension property if there is a domain 9' containing
9 in its interior and a linear mapping 0 of C3(Si) into C.,(f') such that Ou = u
(16)
in 9
110ullc,c9'> < const.
It is easily seen that the ball Ixl < R has the C3 extension property for each s. In fact, we may take (17)
Ou = 1 A,u(R
 k IxI (lx)  R)x)
for R < Ixl < 2R, where k is any integer larger than s and the constants A, are chosen to satisfy k
=o
it
Jk
=1,
0
(This is just an adaptation of the method of Lions described in
the proof of Lemma 9 of Chapter 4.) We also note that by multiplying Ou by a test function in Or' which is identically one in 9, we can always achieve that Ou has compact support in 91'. LEMMA 1.
Assume that 9 has the Ct extension property, t > 0.
Then there is a constant kt depending only on t and the constant in (16) such that (18)
E
kteFrc``
111011C0(Y)
,for0<s
POTENTIAL THEORETICAL APPROACH
219
By virtue of (13), Ct(9) may be considered as a subset of C3(W) for s < t. We show now that the injection of C1(F) into C8(T) is a completely continuous mapping, i.e., that LEMMA 2. Under the hypothesis of Lemma 1, for 0 < s < t every bounded sequence in Ct(V) contains a subsequence which converges in C.(W).
Let {u5} be a sequence in Ct(If) such that Ilu,llc,(,9) M. Since t > 0, the functions u, at least satisfy a uniform Holder condition on W and are thus equicontinuous. Thus by the AscoliProof.
Arzela theorem there is a subsequence (denoted also by {u,}) which converges uniformly on ej, i.e., such that 11 u,  u,llc.o(1) 0 as i, j  oo. Let e > 0 be given and take N so large that \sl(te)/ e
111u, 
4M
'
2kt
for i, j > N, where kt is the constant in (18). Then by Lemma 1, s/(t8) Ilu,

4M
2M + kt . GMlll
for i, j > N. Hence II u:  u; ll c,(w) '
Ilut  u,llco(s) < e
0 as i, j * oo and the
lemma is proved.
We now show that Lemmas 1 and 2 have their counterparts
for the spaces Bk(1). We say that 1 has the Nf extension property if there is a domain NN(W) into NN(V') such that Ou = u
' 7)
const. Ilull.\x(f)
(19)
The ball lxl < R has the Nk extension property for every k and r. In fact, the mapping (17) meets the requirements. LEMMA 3. Let V be a domain having the CJ1_1 and Nx extension properties. Then there is a constant C depending only on k, r, and the constants in (16) and (19) such that
(20)
e
for all j
CE'/(k
L. BERS AND M. SCHECHTER
220
LEMMA 4.
If
satisfies the hypotheses of Lemma 3, then for
1 < j < k every bounded sequence in BA (W) contains a subsequence ,
which converges in B(1).
The proof of Lemma 3 will be given in Appendix II to this chapter. The proof of Lemma 4 is similar to that of Lemma 2 and is omitted. Fundamental Inequalities
5.3.
We shall now state the three fundamental inequalities. It is convenient to write the integral
fix
0(y) dy
in the abbreviated form I yl' * 0(y) = r' * 0. The star represents convolution. If 0 is a Ca, function with compact support, the function r' * 0 is also in C, (for is > n), though its support is not necessarily compact. Our first statement is that of SOBOLEV'S INEQUALITY [1].
If 0 is a C. function which vanishesfor
l xl > R and if q > 1, 0 < A < n, then we have
+1<0 A
1
(a) for
n
(const. depending on R, A, n, q)
Ilr' *
(b) for
1 q
1=0
!n
and any
ql > 1
llr' * 0II1.,,I(R) < (const. depending on R, A, n, q, q1) II0IILp(R) (c)
for IlrA
* 0111.,1(x)
1
1
q,
q
i n
1 >0
< (const. depending on ),, n, q) 11011La(R)
The proof consists of elementary but tricky applications of Holder's inequality and will be given in Appendix I to this chapter.
POTENTIAL THEORETICAL APPROACH
221
Here is an application of the Sobolev inequality. For functions 0 with compact support in I xl < R and having LQ derivatives of first order THEOREM OF SOBOLEV [ 1 ] AND KONDRASHOV [ 1 ].
(q > 1) II 0 II L_l (R'
q1 =
if q < n,
< const. max II D,0 II La(R)
if q > n
II0IILoo(R) < const. max IID,0IILa(R)
nq
nq
and
0 < a < 1  nn
if q > n,
const. max IID,CIIL2(R
q
It is part of the theorem that the lefthand sides are finite.
It is obviously enough to prove the inequalities asserted
Proof.
for the case of C., functions. The first two inequalities follow from the identity (2) and Sobolev's inequality [cases (a) and (c) ; 2 = n  1]. In order to prove the last inequality, we observe that x { h Ix  hits
(21)
x
< const. ihix
Ixi"
for every number a, 0 (x1,
... , x"), h = (h1, ... ,
1
ix
1
ixi"ix
hint1+x
,
a < 1 and any two vectors x h,,), as is easily verified. Now the
identity (2) shows that O(x)

x;
O(x + h) = const.
(IX
so that
I fi(x)  0(x ; h) I < const. IhI°`
(

yl" y'
h,
x' Ix
+ h yI"
ay,
dy
Y IDt'(y) I dy
Ix+h yl"I4 fI I D,0(y) 14y J
IY
y.l,t_i+=
and estimating the last two integrals by the Sobolev inequality [case (a), A = n  1 ; x] we obtain the desired statement. COROLLARY OF THE SOBOLEVKONDRASHOV THEOREM.
If (x) has
L,, derivatives of order k, (q > 1), and if s > 0 is the smallest integer such that q > n/s, then the derivatives of 0 of order k  s are Holder continuous with exponent x for every x such that 0 < a < s  (n/q)
L. BERS AND M. SCHECHTER
222
It is, of course, sufficient to prove the theorem assuming the function 0 to have compact support, since if it did not, one could consider instead the function $ = 0 where 4(x) is a C. function with compact support which is identically I in a given subdomain of For functions of compact support the assertion follows by repeated applications of the SobolevKondrashov theorem. If s = 1, the corollary is just the third inequality of the theorem. Otherwise, we have the repeated inequalities II Dk1c II Lg1(R)
< const. max II Dkc II Lq(R) const. max IID"`l0 IIrg1(R)
IIDxa}1IILgs_1(R) S const. max
IIDs+2
IILgs_2(R)
where Dl denotes the generic derivative of order t° and
ql 
n
nq
q2 =
g
ngl gi n
=
nq 2a n
nq
qs1
= n  (s  1)q > n
Moreover, H,,,R[Dkd¢] < const. max II D'`8+1011 LgJ(R)
since 0 < a < s  (n/q) = 1  (n/qs_1). Combining the above inequalities, we have the desired result. We have already used a special case of the corollary in connection with Hilbert space methods (cf. Lemma 3 of Chapter 3 and Lemma 3 of Chapter 4). The next two inequalities refer to the case where the integral defining the convolution considered is conditionally convergent. We shall call a function K(x)
 ixl )
a singular kernel if cu (x) is positive homogeneous of order 0 (w(tx) _
w(x) for I > 0) and is a C. function for x (22)
w(x) dS = 0
J IxI =1
0, and if
POTENTIAL THEORETICAL APPROACH
223
where dS denotes the area element on the unit sphere. (The smoothness condition on w can be considerably relaxed while the condition (22) of vanishing mean value is essential.) It is easily seen that (22) is equivalent to (22')
K(x) dx = 0
I
R1
holding for all RI and R2 such that O
R2 co(E) dS6fAI r11 . r'i1 dr
r
K(x) d x
.
J I`I=1
JR1
Moreover, if M(x) is any homogeneous function of order 1  n, then M,(x) = (alax;) M(x) is a singular kernel. For (23)
M,(x) dx = I R1
f
R2
ICI=R,
=R1
M( ) _' dSS = 0 R1
by homogeneity. We also note that by taking the limit as R1 * 0 and R2  oo in (22') we see that for any singular kernel
dx = 0
(22 ")
in the Cauchy principal value sense. It is not difficult to verify that for a Holder continuous function O(x) with compact support the integral K * 0 =f0(y)K(x y) dy
exists as a Cauchy principal value:
f 0(y)K(x y) dy = lim CIO
Jr&I>e
0(y)K(x y) dy
The theorems to be stated below will show that the expression K * 0 can be defined for much wider classes of functions. HOLDERKORNLICHTENSTEINGIRAUD INEQUALITY. If O(X) is a C.',
function such that O(x) = 0 for I xI > R then for 0 < x < 1 11K * ¢ II C2(R)
< (const. depending on (o, n, x)
II 0 II C,"(R)
L. BERS AND M. SCHECHTER
224
The proof is given in Appendix I; it is an extension of Holder's classical proof of the continuity of second derivatives of a Newtonian potential of a Holder continuous distribution of matter. CALDERONZVGMUND [1,2] INEQUALITY.
Under the conditions of
the previous theorem and for q > 1 11K * O II
L,(.r)
<_ (const. depending on (o, q)
II
II
L,,t x
This inequality has been proved relatively recently, but has already found many applications to the theory of partial differential equations. The proof is given in Appendix I. We should remark that Calderon and Zygmund also considered more general singular integral operators of the form P(_y)H(x, x y) dy
These play an important role in further applications to partial differential equations (cf., e.g., CalderonZygmund [3]). It follows from the CalderonZygmund inequality that K * u can be defined for u E L?(oe) and that the inequality holds again. Also, we see that the HolderKornLichtensteinGiraud inequality holds for any (b c Q R) which vanishes on Ixl = R.12
We show now that the operator T,;0 introduced above [cf. (7 and 8)] is a bounded operator in the spaces C,,,_x, NV? and B;,, and that its norm can be made arbitrarily small by considering functions with sufficiently small support. Assume that the function O(x) belongs to C,,12(R) and vanishes near lxl = R. Assume also that the coefficients of the elliptic LEMMA B.
operator L considered belong to the space
Then
IITOO11C,_z(R) < const. R"
where the constant depends only on the ellipticity constant of the equation and on bounds on the moduli and Holder constants for exponent x of the coeficients.
12 In fact, it follows from the remark before Lemma 1 that the HolderKorn
LichtensteinGiraud inequality holds for all 0 E C',,(Ri. We shall not make use of this fact here.
POTENTIAL THEORETICAL APPROACH Remark.
225
We stated the lemma for To. But it will be clear
from the proof that a similar statement, with the same constant applies to T?p for any x0, provided we have overall bounds on the
quantities (ellipticity constant of the equation and moduli and Holder continuity of the coefficients). The same applies to Lemmas C and D below.
For the sake of simplicity
Proof of Lemma B.
assume that
n > 3 is odd and that R < 1. Write
iV=(LoL)0=wi+V2 v,(x)
=j _ IPi in [a,(0)  a,(x)}DPO(x) 1pi=m
2(x) =  Y a, 1P1<.111
0 we have that max I
Noting that Hence
I < const. R".
Ix1 `R
IIbpIIC'a(R) < const. R"
Hence, by (14), IIbPDP0IICQ(/C) < const. R"
and by the definition of the norm IIV'iIIca(I{) < const. R"m II'ilcm,a(R)
On the other hand, I IIDp4IICauc) < const. R1 p.in
so that (24)
("a (R) < const. R" m II0I!Cm.a(R)
Now set X = Too, i.e.,
f
/.(x) = Jo(x y)W(y) dy For Ipi < m, it is not difficult to show that DPZ(x) = jDio(x  ),\V( y) dy
II/IICm+,(I(
L. BERS AND M. SCHECHTER
226
Thus II DPxII co(R)
const.
IlrmnIPI
* ItVI IIC0(R)
const. RmIPI I10IC,(R) < const. R"IPI
and hence RIP' IID9xII co(R) < const. R" II0IIC,.(R)
(25)
If IpI = m, we claim that
f
DPx(x) = DPJo(x y)V(y) dy + const. ip(x)
(26)
We first prove (26) for W(x) in C1(R). If IpI = m  2, then by partial integration D,DkDP f Jo(x y)V(y) dy = D, f DkDPJo(x y')V(y) dy
= f DiDPJo(x y)DkV(y) dy f y)V(y) dy = lim f IzvI>e DkD,DPJo(x D,D"Jo(x Y)V(Y)Yk dy  lim f IzYI=e = f DkD,DPJO(x y)V(y) dy + const. W(x) = D, DPJo(x y)Dkv(y)
dy
IO
e, O
where yk is the kth component of the unit normal to the sphere Ix  yl = e. Next we note that DP J (x) is a singular kernel for
IpI = m. This follows from (23). Hence by the HolderKornLichtensteinGiraud inequality and (26) we have (27)
IIDPxIIC2)R) < const. IItVIIC(R)
By completion it follows that (27) [and hence (26)] holds for y, c C. (R).
Combining (24), (25), and (27) and recalling the definition of the norm we have IIXIICm+i(R) < const. R" IIOIICm+z(R)
and the lemma is proved.
POTENTIAL THEORETICAL APPROACH LEMMA c.
227
Let the function 0 belong to B7,(R) and vanish near
I xI = R. Assume that the coefficients of the elliptic operator L considered are continuous (these conditions can be weakened considerably) and that
q > n. Then IIT00IIB(R) < a(R) 11011 mIm
where a(R) denotes a junction depending only on the ellipticity constant of
Lo and the bounds and moduli of continuity of the coefficients of L.
Moreover, a(R) , 0 as R  0.
Using the same notation and conventions as before, we have that Proof.
Rmn/q
11IP111L°(R) S max IlbDllco(R) 
Rmn/q I
IID9911L°(R)
IpI =on
< o(l) II/IIBm(R)
and Rmn/v
const. max Ilapllco (R) IID9
11tP211L (R) °
x
JjxI
IPI
IIc0 (R)
cont. R
dx
o(1)
J
so that (28)
R"'q IIWIII.°(1(>
C 0(1)
Applying the CalderonZygmund inequality to (26) we have for IPI =m (29)
RmnI4 IID1%11L°(R) C o(1)
For 0 < IpI < m we estimate q > n. We have that
II Dny II C,, (R)
using the assumption
I D"J (x) I C const. lxI mnIDI
so that for Ixi < R ID"A
const.
IrmnIDI
* Iv'l
I
1 /q'
< const. 11+P11L°(R)
)( 1
Ix
l IvI
< const. RmIvlnlq
IIWIIL,(R)
yl(mnIDI)q' dy)
L. BERS AND N1. SCHECHTER
228
where q' = q/(q  1). Hence (30)
const. R""I" IIl'IILQ(R)
RI PI
o(1) II0IIJ (R)
by (28). Inequalities (29) and (30) give the lemma. LEMMA D.
If the coefficients of the elliptic operator L are continuous,
then for each q > 1 IIToOlI
(R) c a(R) II0II.\m(it)
for all S e NN,(R) which vanish near jxl = R. The function a(R) > 0 as R  0 and depends only upon q, the ellipticity constant of Lo and bounds and moduli of continuity of the coefficients of L.
The proof of Lemma D is very similar to those of the preceding lemmas and is omitted. 5.4.
Local Existence Theorem
As the first application of the tools developed above we shall prove an existence theorem in the small. THEOREM 1.
Let L be an elliptic operator of order m with Holder
continuous coefficients and f a Holder continuous function. In a sufficiently
small neighborhood of a point, say of the origin x = 0, there exists a solution u of the equation Lu = f having the following properties: (i) u has Holder continuous derivatives of order m. (ii) u and its derivatives up to order m  1 have at the origin prescribed values. (This solution is,
of course, not unique.) Proof.
We first note that
1 on C,,(R) functions which
vanish on Ix) = R. For if 0 is such a function, we can find a sequence (0v} of Cr functions which vanish for Ixl > R such that
'IIC2(R) ' 0
asvkcc.Nowby(5) and by (26)
LOSOO, = 0.
r
LoSoc,(x) =J LoJo(x  y)O,(y) dy ± const. ¢v(x)
POTENTIAL THEORETICAL APPROACH
229
Hence by the HolderKornLichtensteinGiraud inequality in
LOS00,  LOSOO
C. (R)
giving LOSOO = 0
Let fi(x) be a Co, function which is identically one in IxI < 1/2,
vanishes for IxI > 1. Then by (15a) II
S(RIx)
11C'm+x(R) = II y(x) 11Cm+a(l)
Set
Mou(x) = Toq(x)u(x)
where n (x) = (R'x) . Then by Lemma B, Mo is a bounded linear operator which maps C. ,(R) into itself. Moreover its norm 11Moll = l.u.b. {IIMou11Cm+x(R)/Ilullcm+«(R)}
less than const. R« II fi(x) Ilem+«ci) and hence can be made arbitrarily small by taking R small. If IIM011 = e < 1, then the "geometric series" 1 + M0 + Mo + ... is dominated by 1 T e + e2 ; ... and is therefore easily seen to converge to a bounded operator is
x
(1 Mo)1=IMu j __o
which maps Cm+x(R) into itself, satisfies the relations
(1  MO)'(1  M0) = (1  Mo)(1  Mo) 1 = 1 and has norm II(
1M
0)1II
OIIs
<1 +e+e2
j=0
1
1e
Set
g(x) = I Y,,xp 1A 11,
where the yp are constants. Clearly Log = 0. If
u = (1  K) 'g we claim that Lu = 0 for IxI < R/2. Indeed we have
uMou=uS0(LoL)i7u=g and hence Lou  LOSO(L0  L),qu = Lou  Loqu + L,1u = 0
L. BERS AND M. SCHECHTER
230
Since 77(x) = 1 for Ixl < R/2, u is a solution of Lu = 0 near x = 0. Also
Ilu 
IIM01I
s IIMoII 11(1  Mo)'1I IIgIIc,.+2(R)
<
IIgIIc.+,(R)
1
Recalling the definition of the norm II IIC,a+,,(R)' we see that this implies that
IDiu(0)  YPI s Ce' I
IPIsm1
IyDI
for IpI < m  1
where C is fixed constant and e' can be made arbitrarily small by making R sufficiently small. Now denote by u(PO) the solution just obtained for the following choice of the constants Y,,: yDO = 1, y,, = 0 for p : po. Every linear combination
u = I cDu(P) 1PI _M 1
is a solution of Lu = 0. We want to determine the constants c,, so as to satisfy the equations I cDDQu(P)(0) = YQ (IqI < m  1) IPI<m1
i.e., as many equations as there are unknowns. We know already that by choosing c,, = yp we can satisfy these equations approximately (with an error not exceeding Ce'). Hence, by a wellknown theorem in linear algebra, the equations are solvable.
The theorem is now proved for f = 0. In order to settle the general case it suffices to exhibit one solution of Lu =f which belongs to C,n+,,(R). (We assume that f c C,,(R).) By our reasoning above it suffices to exhibit a solution u of the equation
u = Mou + S0f But if IIMoII < 1 such a solution is given by u = (1  Mo) 'Sof 1. THEOREM 2.
Assume that the coefficients of the elliptic operator L of
order m are continuous and f is a function belonging to L q > 1. Then
POTENTIAL THEORETICAL APPROACH
231
there exists a solution u of the equation Lu = f which belongs to the space NN(R) for small R and satisfies condition (ii) of Theorem 1. If q > n, we can find such a u in B qm (R)
.
The proof is the same as the one given above, except that Lemmas C and D are to be used instead of Lemma B. The continuity assumption on the coefficients can be considerably weakened. An interesting corollary of Theorems 1 and 2 is the fact that by
introducing new dependent and independent variables an elliptic equation can be transformed, locally, into a form in which the (undiferentiated) unknown function and its first derivatives do not appear explicitly.
In particular, every homogeneous second order equation can be written, in the small, in the form
Iail
a2
=0
In order to prove this let 0o be a solution of Lq = 0 such that 00 = 1 at x = 0. In the neighborhood of the origin introduce y, _ (0/00) as a new unknown function. Then tp satisfies an elliptic equation Liy, = 0 and since y, = 1 is a solution, the coefficient
of y, in Li is zero. Next, let V,, j = 1, 2, ... , n, be a solution of LIV = 0 such that 1
any;
aXk
I
z=o
0
if j = k if j Ok
and y,, = 0 at x = 0. We may introduce ; = V, as new independent variables and obtain an elliptic equation L2W = 0. This equation is satisfied by every linear function. Hence the coefficients of y, and all first derivatives in L2 are zero.
5.5. Interior Schauder Type Estimates
Some of the most important applications of potential theoretical methods are a priori estimates of solutions of linear elliptic equations of the type given by Schauder for second order equations and then extended by other authors to higher order equations and systems.
L. BERS AND M. SCHECHTER
232
Schauder assumed the equations to have Holder continuous coefficients, but using the CalderonZygmund inequality the method can be extended to continuous coefficients and even to more general situations. In this section we prove two theorems on socalled "interior" estimates. Let L be an elliptic operator of order m with coefficients in C«(9). Let 9 be a bounded domain contained, together with its closure, in .9. Let u belong to C,,,+«(W) . Then, for every compact THEOREM 3.
subset To e (31)
Ilullc.,«(T0) S C(ItLullc«(,,) + Ilullco(f))
where the constant C depends only on m, W, WO, the ellipticity constant of L and bounds for the C«(Wl) norms of the coefficients of L.
If
and WO are given, we can cover WO by a finite number of open balls whose closures belong to W. Hence it suffices to prove
the theorem for the case where 9 and 90, are concentric balls of small radius. Without loss of generality we assume that the center is at the origin and prove Theorem 3 in the form (32)
C 1  Rr)' (IlLullc«(R) + Ilullco(R))
for 0 < r < R and R sufficiently small, where T The proof is based on the preliminary estimate n«
(33)
IIUIIC..+«(R,) < Cl
1  RZ
(IILuIIc«(R2) + Ilullc.,T«(R2))
holding for all Rl and R2 such that 0 < Ri < R2 < R. Assuming (33) for the moment, set
A = sup
0
r'
R Il u 11 c
There exists an R1, 0 < Rl < R, such that A<2 1
R') IIUIIcm+«(R,)
POTENTIAL THEORETICAL APPROACH
and using (33) we have, for R1 < R2 < R Rlnaa{IILuIIC(R,) <2(1 RR1)TCi(I
A
+ IIUIICm,+a(R2) 2
<

2C1(1
m={IILuIICa(R)
1  R2)
R11*I
+ lUll Rl
<
ina
 W 1  R2)
2C1(1
[by (lib)]
_l+acR2)}
{IILuIIC«(R)
(34) + E IIUIIC,n+a(R2)
<

2C1(1
Km+aE(m1+a)

RRi1T(1
(by Lemma 1)
II U II C0(R,)}
RR_1)maE(1

RR21A
21
Rilr
Ri
m a
IILuIIC«(R)
RI (1
R2) lm_a 811(1T2)
xr»+a
F(mla) IIUIIC0(R)
/
Set d = 1  R1/R and choose R2 and E as follows : R2 1R
b
2,
e=
22m2*6m)XjC
1
Then
2<1R
0
0<e<1
2
and
2C11R
( so that by (34) A S 4C1(1
 R1
+ 4C1(I
'Rl /
(1

R1 =
R) (
(1_Ti)e<
2
mallLullCa(R) R1)
R2 Y R1'
1
 mz
R 2)
km+x
E(m1+s)
< 4C12(m+(x)(=+2) km+z (llLullca(R) + IIUIIC0(R))
IIUIIC0(R)
L. BERS AND M. SCHECHTER
234
It remains to prove (33). We choose R so small that (35)
IIToOllc.+«(R.) < k IIOIICm+,(R2)
for all R2 < R and 0 with support in Ixi < R2 (cf. Lemma B). Let cu(t), 0 < t < 1 be a fixed C. function such that w(t) = 1 for t < 3 and w(t) = 0 for t > I and set, for 0 < Rl < R2 for IxI < Rl
1
for R1 <x
R)
2
2
One verifies at once that there is a constant Co such that (36)
II IIC.+«(R2) < CO(I \
 Ri 2
4u. Then the support of 0 is in Ixl < R2 and by
Now set Lemma A
= Too ± SOLO
By (35) this gives II O II Cm+«(R2) < 2 it SOLO II c,.+0C(R2)
Now by (26) and the HolderKornLichtensteinGiraud inequality IID'SOLOIIC(R2) < const. IILOIIC«(R2)
for IpI = m while for IpI < m lrmnIPI
ID'SoLOI S const.
* ILOI
< const. Rm1111 JLcl
Hence (37)
IISoLMIICm+«(R2) < const. Rm IILOIIC«(RE)
But
Ld=CLu+N
where N is a linear combination of derivatives of u up to order m  1 multiplied by derivatives of C up to order m, a derivative of u of order k being multiplied by a derivative of of order m  k. Hence Rm II J
it
C«(R,)
< const.
II
II
Cm+Z(R2) II u II Cm_,+«(R2)
POTENTIAL THEORETICAL APPROACH
235
and therefore R", IIL0IIc«(R2) < const.
(IILuIIc,,(R2) + IIUIIcm_1+a(R2))
Combining this with previous inequalities we obtain const. IIVCm+.(R2) (IILuIIcx(R2) + Ilullcm_,+z(R2))
Since 0 = u for IxI < Rl Ilullcn+«(R,) S
const. II IIc,p+a(R2) (IILukkc,,(R2) + IIUIIC,._j+,,(R2))
which is the desired result in view of (36). We consider next equations with continuous coefficients. In the same way as Lemma B implied (33), Lemma C leads to the preliminary estimate (for q > n) (38)
IIUIIR",(RI) <
Cl(i
R
Yin
(IILuIILQ(R2) + IItIIRm1(R2)) 2
From here we obtain, exactly as before, the inequality II1IIRm(t)
C(1
 R)
m2
(IILuIIL,,(R) + Ilulle"(R))
and thus we have Let L be an elliptic operator of order m with continuous coefficients defined on the closure of the bounded domain 9. Let 9o be a THEOREM 4.
compact subset of K If u belongs to Bn, (0), with q > n, then IluiIIim(fgo) s C(IILuIIL,(T) + IIUIIe0(1))
where the constant C depends only on m, !F, To, the ellipticity constant of L, and bounds and moduli of continuity of thecoefficients of L.
We do not state a corresponding theorem for the Nnorm, since we shall not need it. 5.6.
Estimates up to the Boundary
There exist also estimates "up to the boundary." We state here the second order case for the Dirichlet problem which is due to Schauder [1,2]. The corresponding result for equations of higher
L. BERS AND M. SCHECHTER
236
order and more general boundary problems has been proved by AgmonDouglisNirenberg [1]. An (n 1) dimensional hypersurface in n space is said to be of

class Cx +z if every sufficiently small piece of it can be mapped into
a piece of a hyperplane by a transformation of coordinates y(x) with positive Jacobians, the functions y = ( 31 , . . . , y,(xl,
... ,
being of class Ck
.
Let T be a bounded domain with a boundary hypersurface W of class C2;a. Let the coefficients of the second order elliptic operator L belong to C,,(V). Let u be a junction belonging to C2+,, (T) . THEOREM 5.
Then (39)
Ilull(.2+M(1) < C(IlLullca(1) + Ilullc2+Yc%4) + Ilullcoc9))
where the constant C depends on p, the ellipticity constant of L and bounds on the Ca(lf) norms of the coefficients of L.
The method of proof is similar to the one described above, except that one must also use explicit formulas for solving the Dirichlet problem for Lou =f in some standard domain. For the second order case, this is relatively simple and we sketch the procedure. We first note that may be covered by an interior subdomain and a number of small "boundary patches." By hypothesis, each
boundary patch can be mapped in a onetoone way onto the semisphere Ixl < R, x,, > 0 with the boundary being mapped into
the hyperplane x = 0. Moreover, the mapping and its inverse have Holder continuous second derivatives. One then observes that by subtracting an appropriate function from u, it suffices to prove (39) for functions which vanish on 47. Thus it suffices to prove (40)
Ilullcp+a(s,) < C(1
r)7 R
(IlLullca(ER) +
for r < R and functions u which vanish on x = 0, where 1. is the semisphere Ixl < r, x > 0. This is done, as in the previous case, by proving (41)
Ilullc+a(='R1) < C,1(1
R R2)
2z
(IILuile2(rRI) + Ilu11c,_=(ER:))
for all Rl and R2 such that R1 < R2 < R.
POTENTIAL THEORETICAL APPROACH
237
In order to prove (41), we first note that
f
W(x) = Lo W(y)Jo(x y) dy
holds for W E Ca(R) vanishing on Ixi = R (cf. the proof of Theorem
1). Now if O(x) is a C,., function in >;R which vanishes on x = 0 and near Ixj = R, we can extend it to xn < 0 as an odd function of xn. Thus O(x) will be a C1(oo) function vanishing for Ixi > R and satisfy
0(x') = 0(x) where x' is the image of x under reflection with respect to the plane
xn = 0. We can therefore write
J (y)Jo(x y) dy =U0(y)Jo(x y) dy n>o
+f where
n
0(y)J(x y) dy
Jo(x,y) = Jo(x y) Set
=f R`o 0(y)Jo(x,y) dy
 Jo(x y')
S0'0 (X) = f,vn o0(.y)Jo(x,y) dy
To = S' (Lo
 L)
Then we have, by partial integration,
0=
S0L
provided 0 vanishes on x = 0. The proof now proceeds as in the "interior" estimates. One shows that an inequality of the HolderKornLichtensteinGiraud type holds for kernels of the form D"J (x) for Ipl = 2, i.e., s const. II0IIC%(
R)
IpI = 2
The proof then goes through with little change. 5.7. Applications to the Dirichiet Problem
In order to apply the Theorems 3 and 5 one must keep in mind that the constants in these estimates depend only on the domain
L. BERS AND M. SCHECHTER
238
considered, the ellipticity constants of L and the Ca norms of the coefficients of L but not on the particular operator L considered !
As an application of the Schauder estimates we will solve the Dirichlet problem for the second order equation 2
(42)
Lu =
a,,(x) ax ax i
+ I a,(x) ax
+ a(x)u =f(x)
in a domain T satisfying the conditions of Theorem 5. We assum that f and the coefficients of L belong to C2+x M and that a < 0,
so that the maximum principle is applicable. We consider first the boundary condition (43)
u=Oon1,
0eC2+(V)
We assume also that we know how to solve [in C2+ac(g)] the corresponding Dirichlet problem for the Laplace equation. (This can be done, for instance, by the method of integral equations or the methods of Chapter 4.) The proof (Schauder [1,2]) is by the continuity method. Assume
first that g  0. For 0 < t < 1 set Lt = (1  t) A + tL and denote by T the set of those values of t for which the Dirichlet
problem Ltu =f, u = 0 on 0 has a solution in C2.+2(T) for all f in Ca (T) . We already know that t = 0 belongs to T. We shall show that r is open and closed. This means that r is the closed interval 0 < t < 1, so that the homogeneous Dirichlet problem is solvable for t = 1, i.e., for L, = L. Proof That T Is Open.
Assume that to belongs to T and denote
the solution of Lt u =f, u = 0 on
by Mf. (This solution is
unique by the maximum principle.) Clearly M is a linear mapping of Ca (q) into C2+ (T) and by Theorem 5 we have that (44)
IIMf
C IIf II Cr(y)
The equation Lto+,u =f may be written in the form
Ltou =f + E(O  L)u so that a solution of the equation (45)
u = EM(L  L)u + Mf
POTENTIAL THEORETICAL APPROACH
239
would be the desired solution of the homogeneous Dirichlet problem for Lto+Eu =f.
Consider the linear operator N = M(i  L). Clearly 0  L maps C2+«c91> into C
)
and
II (o  L) u ll c« (9) < const. II u II c:+«(V)
so that N maps C2+x (T) into itself, and by (44) IINII = l.u.b.
IINuIIra+s+«(g)
Ilullc
Equation (45) is equivalent to
(1  sN)u = Mf and if IEI < K1 it has a solution
u = (1  sN) 1MF = I E'N' Mf 0
Hence to + e belongs to r if lel is small enough. Proof That r Is Closed.
Assume that t j = 1, 2,
.
.
.
,
belongs
to r and tf * to. Let 0 be the solution of L,.u""> =f, u(') = 0 on #. By Theorem 5 IIUW IIc:+«(g) < C1
(a constant independent of j )
By Lemma 2 we may assume, selecting if need be a subsequence,
that u(' converges uniformly, together with its derivatives of orders 1 and 2 to a function u. Clearly u belongs to C21 «(g), vanishes on the boundary and satisfies the equation Ltou = f. Hence to belongs to r. In order to solve the nonhomogeneous boundary value problem (43) we denote by h the harmonic function satisfying this boundary condition and solve, as we now know how, the equation
Lv=f,
f=fLh
under the boundary condition v = 0 on 0. Then u = v + h satisfies condition (43) and the equation Lu = f. Consider next the boundary condition (46)
u = 0 on (,
0 E Co(9)
L. BERS AND M. SCHECHTER
240
We construct a sequence of functions 0; defined on
j
II0;
such that
 OGc'o(4)  0
(For instance, take the 0; as polynomials.) Let us be the function in C,,. (6) determined by the conditions Lu, =f, u,
on.
By the maximum principle u; converges uniformly to a function u. Thus I'Uj  uIlco(rs)
(47)
0
so that u c Co(11) and u = cb on V. On the other hand, by Theorem 3 for a compact subset 10a e G°, II u;  u, II c$+a(vo) S const. 11u,  ut 1, ca(1o)
the constant being independent of i, J. It follows from (47) that u has Holder continuous second derivatives and satisfies the
equation Lu =f. In a similar way, using the estimates in Theorem 4 and approximating the coefficients of (42) by nice functions one can
show that if L has continuous coefficients, the equation Lu =f has solutions with continuous first derivatives and generalized LQ derivatives of second order in every subdomain. 5.8.
Smoothness of Strong Solutions
We now give a proof of the theorem on continuous differentiability of strong solutions (cf. Section 3 of Chapter 1). Assume that f and the coefficients of L satisfy Holder conditions with exponent x. Let u be a function having strong derivatives up to order m in
L. for some r > 1 and suppose that Lu =f almost everywhere. We shall prove that the mth order derivatives of u satisfy a Holder condition with exponent x and hence u is a classical solution of Lu =f. Since the theorem is local in nature, we need only to prove it in the neighborhood of some point, say the origin. Let fi(x) be a Cx function which is identically one in Ixl < 1 and vanishes for lxl > 2. Setting r7(x) = (R'x) we have by (15a) (48)
1117 IIc,(2m = II 0C,(2)
POTENTIAL THEORETICAL APPROACH
241
for any s. For v c Nn(2R), r > 1, set Mov = T077v
Now by Lemma B, (14) and (48), we have for v e Cm+,,(2R) II T0nv I I Cm+:c (2R)
< const. R' II v II Cm+cx (2R)
Similarly, one sees by Lemma D, that for v c NN(R) 0(1) IIvllAm(2R)
as R  0. Hence we may take R sufficiently small so that (49)
Il Mov ll Cm+a (2 R) `= z II V I1 Cm+%(2R)
and
(50)
11M0vll \m(2R) <
Ilvll
Now let Ri be any positive number less than R and let 0 be any
test function in lxl < R which is identically one in lxl < R1. One easily verifies by means of the CalderonLygmund inequality,
that Lemma A holds for functions in Nm(2R) having compact support (for a similar argument see the proof of Theorem 1). Thus cu = M0 u 1 S Lou since Ou = Oqu. Now by (50), 1  M. has a bounded inverse in Nm(2R) (cf. the proof of Theorem 1). Hence ou = (1  MO) 1S0LOu
Now Lou = Of 4 Liu, where L1 is of order less than m and has
Holder continuous coefficients. Now if r > n, we see by the SobolevKondrashov theorem that L1u satisfies a Holder condition
with exponent fi < 1  (n/r). If fi > x, we have Lou E C,,(2R). Hence, by the HolderKornLichtenstcinGiraud inequality S0Lou is in Cm+ (2R) . But by (49), (1  M,)1 maps Cm+.x(2R) into itself and hence Ou is in Since 0 = I in Ixl < R1, we have u E C,,,+2(R1) and our assertion is proved. If j3 < x we know that Lou is in CC(2R) and hence SOL6u is in C,,,_g(2R). By taking

R sufficiently small we know that (1 M0) 1 maps C,.+,, (2R) into itself. Hence u r= Cml,3(R,). But this implies that Liu is in
L. BERS AND M. SCHECHTER
242
for any
C,,(R1). Hence by our reasoning above, u E
R2

Kondrashov theorem asserts that Llu E L,1(2R), where r1= nr(n r)1. Hence LSu e Lrl(2R). Applying the CalderonZygmund in
equality we see that S0Lciu e Nm(2R) into itself. Hence 6u E N», (2R) and u E A,, (R1) . If rl > n, the argument above then completes the proof. Otherwise, we note that L1u E Lr2(R1), where r2 = nr1(n  rl) 1 = nr(n  2r). From this it follows that u e N,., (R2)
for R2 < R1. If k is the largest integer such that kr < n, we repeat the procedure k times to obtain u e NN(Rk), with rk = nr(n kr) > n. We then know how to proceed to the final result.

Appendix I. Proofs of the Fundamental Inequalities
We shall outline proofs of the three sets of inequalities stated in Section 3 of Chapter 5. 1. Sobolev's Inequalities.
The proof of (a) is trivial. In fact, by
Holder's inequality IrA
*
('
If Ix y1'o(y) dy
ylaq dy)
Ix
I
IyI
S const. R("17') 2
l f q'
110 II L0(f)
where
q'=
q
q
1
n ,1=nI 1 n) >0
and
q
q
We base the proof of (c) on a onedimensional inequality of Hardy and Littlewood [1] (cf. also Zygmund [1]) which states that for f c Lq,(  oo, oo) and g e L,, (  oo, cc) x
 111f(s)g(t) ds dt fHs x
xx
< const.
'
If (s) I'1 ds) 
when 1
q>I'
p>I'
I +>I,
p
q
/F
14
l .j
Ig(t) 111 dl I
x
2I
/
POTENTIAL THEORETICAL APPROACH
243
In order to prove (c) it suffices to show that I(ri * 4, )I < const. II¢IILQcx)
(1)
II
IILQI(Go)
for all 0 and p with compact support, where qI =
ql f
(qi I).
We make use of an observation due to Du Plessis [1] that
IxIa < const. rl
(2)
Ix;1a1fl
This follows from the fact that an arithmetic mean is greater than the corresponding geometric mean. Thus
fix; n j i
?IIIxjI21"
j=1
which yields (2). Employing (2) we have I (r' * 4', V) I < const. I j I xj  yf l 'In 10(x) I I v (y) I dx dy. j=1
Setting p = qi and It = A f n, we apply the HardyLittlewood inequality to the integrations of x1 and y1. We then reapply it to each pair of integrations x5 and y; eventually obtaining (1). In order to prove (b) we note that if 0 is in Lq,(R), then it is in L,,(R) for all p < q. Let q2 be so large that q, < q2 and 1 + I < 1
(3)
Setting 1
_
P
we have q >p > 1 and
I
q2
llr_a * 4'IIr,Q,(R) < const.
=
1
q2
1
p
+
1
+q A
 1. Hence by (c)
n
n((i141)(1/q2)] 1jrA * IIc.Q=cx)
< cont.
cont. R11171
IILQ(R)
Remark. Note that under assumption (3) we have proved Ilr' * for case (b).
const.
R"QE
II4JL9(R)
L. BERS AND M. SCHECHTER
244
2. The HolderKornLichtensteinGiraud Inequality.
We first note
that by (22") of Chapter 5,
f
K * 0 = [O(y)
(4)
 O(x)]K(x y) dy

Let x' and x" be any two points in IxI < R and set 6 = Ix'x"I.
If ,=K*0,wehave w(x')
 '(x")
=f {[O(y)  O(x')]K(x' y) =I1+I2
 [O(y)  sb(x")]K(x" y)} dy

where Il is the integral over the sphere 01: ly x'I < 26 and I2 the integral over the domain 02: ly  x'I > 26. We note that if
y2 isIx"inyl 02 and x' is such that Ix'  x'I S 6, then Ix'  yI < s 4 Ix' yl. Now II11
< H.,x[0] fa { IK(x' y)I Ix' yl' 1
(5)
cont. &H«,R[f]
+ I K(x" y) I Ix"  yi"} dy
Also, (6)
 O(x')]K(x' y) dy
I2 =fo [0(x")
+f !2
[O(y)  O(x")][K(x' y)  K(x" y)] dy
Now by (22') of Chapter 5
K(z) dz = 0 L>20 and hence the first integral on the right hand side of (6) vanishes.
Moreover, by the theorem of the mean, there is a point x' between x' and x" such that
IK(x' y)  K(x" y)I <
const. 6 1x
_yln+j
Thus ify is in 02 (7)
I K(x' y)  K(x" y) I <
cont. 6 Ix,
y19+1
POTENTIAL THEORETICAL APPROACH
245
Employing (7) we have II21 S const. 6H«,R[9] (8)
f'_ ly 
x'l"nI dy
< const. 6"Hx,R[0]
The result now follows from (5) and (8).
Our proof is based on a
3. The Calder6nZygmund Inequality.
classical theorem of M. Riesz [2] (cf. also Zygmund [1]) which asserts that for q > 1 f f(t) dt q dT S const. f 00 1 f(t)IQ dt Joo j_ tT oo °°
I
Clearly, Riesz's theorem is just the CalderonZygmund inequality
for n=1. Assume first that the singular kernel K(x) in n space is odd:
K(x) = K(x). We carry out the integration in the integral defining K * 0 first along the radii and then along the surface of the sphere. Thus
K*0=
fo(y)K(x y) dy =
fo(x  z)K(z) dz
= lim f _ico() dSS f O(x 00
rn1 dr
eI
where r = IzI. In view of the oddness of K(z) [and hence of co(z)] the integrals along two oppositely directed radii can be combined into a single integral. Hence K*
=
and
I K* l a dx < const.
f
00
aj (E) dSf
f;i=1 dSJ J
O(x  r)r1 dr 00
f(x  r) r'1 dr
I
dx
Since r1 is a onedimensional singular kernel, we have, by Riesz's theorem, that for fixed
r
I (x + t  r )r1 drl
J
a
dt < const.
f I0(x + t )IQ dt
L. BERS AND M. SCHECHTER
246
Integrating along axes perpendicular to
l
gives
Q
dr
O(x 
fo.
dx < const.J I0(x) I,, dx
which, when combined with the preceding inequality proves the theorem for odd kernels.
If K(x) is a singular kernel, let us denote by K the operainto the function K * 0. We tor which takes the function consider the particular singular kernels (Riesz kernels) K,(x) = cxjxln1, i = 1, 2, . . . , n and note that for an appropriate choice of the constant c, 1Q2 + ... + ten = 1 = identity. To see this we observe that K. (X) = c1
a
i = 1, ... , n
Ixl'n' ax2
Thus by (4) ci
f 10 (Y)  0 (x)] ax Ix yi1" dy
Now by partial integration we have for R1 sufficiently large,
f
i
[0(y)
 fi(x)] ax Ix Y11 dy ,
a
JF<,vzl
ay
I
x 'nd yl y + ( x)
Ix
vZ,=Rj
'
J O(x)]
=d Y1" X'
_ yly dy
The first boundary integral vanishes identically while the second tends to zero as r  0 by the continuity of 0. Hence
10 = ci
aa(3) y=
Ix
_yl'" dy = ci Izlin
z)
ao (X
dz
,
Hence (9)
iV = clr'" * OX =
C1
(rin
*)
for test functions,. Now by Sobolev's inequality and the usual limiting procedure, we see that (9) holds for a Cam, function having first derivatives in Lq(oo) for some q < n. But the same inequality
POTENTIAL THEORETICAL APPROACH
247
tells us that v = X,0 is such a function when 0 is a test function. Therefore, assuming for the moment that n > 2, we have
f
Ci
Izlln
aR1 (x + z) dz ax,
J
c1J IZI1n axt  J IUIin a0(x axz, + u) du dz
+ z + u) du dz _ C f Izlln lulln a2o(x ax2 Izlimyzlln a2o ax+ y) dv dz
= GIJ
_c
f
F(v)
a2¢(x
+ v)
ax!
where F(v)
=f
Izi1n Iv  zIln dz
Clearly F(v) is positive homogeneous of degree 2  n and invariant with respect to rotations about the origin. Hence F(v) = const.
IvI2n
This means that X20 ,
Therefore
a2o (x
const.
+ v)
dv
J1v12n
f
1X?¢ = const. IvI2n 0O(x + v) dv which equals const. 0 by (1) of Chapter 5. When n = 2 the interchange in the order of integration above
is not justified by the behavior of
Izlln Iv  zI1n at infinity.
However, by being slightly more careful we can arrange things so that only finite integrals are interchanged. Assume that O(x) vanishes for IxI > Ro. Then by (9) we have (10)
a
Cl ai
(r1 * K,O)
L. BERS AND M. SCHECHTER
248
But
r1 * 1Q, = c1 lim fIzj R. x
J'vl < Ro
a0
= c1 lim
(11)
R x flyl < Ro ay:
cl f
Ix
IzII
lim
f
+z
dy dz
a1',
IzI1 Ix + z yII dz dy
=I
zI
a
IzI1 Ix + z  yI 1 dz dy
lyl
Now consider x, y, z as complex variables and set t = x y, Z = z/t = pe'o. Then
f I
IzI1 Ix + z
f2,,f,R/Itl Itl2 p dp dO
y11 dz
ItZI Itz  tI
U
J'"Jo R/IrI
dp d
I pe'o  1
o
Hence
f
IzIIIx+z
ay, fzI
ylIdz=
f
2,r
U
dO
R e io
R
f
1
l
2
Itl
t;
f21r
t2 211
de
eto_Itl R
and
Jim
Jfzl
ay,
z yI1 dz = Tr t=2
Substituting back in (11) we have
f
r1 *K,O = const. o(y)
xt Y,
Ix yl
dy
f y3 const. f a' log Ix  yI dy ya
const. 0(y) a log Ix  yI dy
Hence by (10) L20
= const.
0 log Ix yI dy
fay,
t, ItI3
POTENTIAL THEORETICAL APPROACH
249
and R2)
f
= const. 0# log Ix yl dy
2
= const. 0 by (1') of Chapter 5, and the assertion is proved for n = 2 as well. Now let K(x) be a given even singular kernel [K(x) = K(x)]. We have that k = Z (IMI)1e=. If we can show that R i may be written as M where M=(x) is an odd singular kernel, then (L1 ,) and Xj will both be bounded in the L, norm by the result obtained above. This will then imply the CalderonZygmund inequality for K. Since every singular kernel can be written as the sum of an even and an odd kernel, the proof will be complete. It therefore remains only to show that kki = M=, where M,(x) is an odd singular kernel. Now R.0
= cifIx Y1'" aa(y)
dy
y=
and therefore
ciJ K(x _)f aazz) 11.),  Z11  Ix  yll'=] dz dy = ci
r I
az
M(x
 z)
dz
where (12)
f
M(x) = K(x y)
[Iyi'n  Ixlin] dy
Obviously, M(x) is positive homogeneous of degree 1  n. Hence, for x 0 0, it will follow from (23) if we can show that it is in
of Chapter 5 that M=(x) = (a/ax;)M(x) is an odd singular kernel. In addition, for Ri sufficiently large, we shall have R=>lxzl>a
M(x  z) aazz) dz =
M, (x  z) [0(z)  0(x)] dz
Rl>Ixzl>a
 z  (x) JIxzh=Ri M(x  z) x' dz Ix zI _
fixZI8
M(x  z)[O(z)
 fi(x)] X,
z
Ix
zI
=
dz
L. BERS AND M. SCHECHTER
250
The first boundary integral vanishes since K(x) (and hence M(x)) is even. The second tends to zero as e > 0. Thus it will follow that
9'0 = C,
[O(z)  O(x))M1(x  z) dz
and the proof will be complete. It therefore remains to show that M(x) is in C,,, for x 0 0. This is immediate if one writes (12) in the form M(x)
=fzj< (Ix 
IR,
Z11n
 Ixlln)K(z) dz
Iyiln K(x y) dy
zvl>R,
where R1 is some fixed positive constant less than I xJf2. The function Ix  zIln is in C" for IzI < R1 and vanishes for z = 0. Hence we may differentiate under the integral sign. Since K(x y) is in C., for Ix yi > R1, the same is clearly true Ixlln
for the second integral as well. This completes the proof. It is obvious from our proof how one may relax considerably the smoothness assumption on &)(x).
Appendix II.
Proofs of the Interpolation Lemmas
We now give the proofs of Lemmas 1 and 3 of Section 5.2..
Assume that 9' is contained in some cube with sides of length 2R. Since we may take Ou to have compact Proof of Lemma 1.
support, we may assume that 9' is the cube. Thus it clearly suffices to prove IIOIlc,((l) <_ E IIOIIc,(
') +
ktE8/t8
for functions 0 c C,(9'). Setting M. = II O II
we first note that for 0 < r1 < r2 < r3
the inequalities M1.1
for all 0 <E < 1
< EM,, +
A1,= < EAIra
+
'2 MO
for all 0 <e < 1
POTENTIAL THEORETICAL APPROACH
251
imply
Mr, < PM,,, +
k12Pr,,r2 r1Mo
k23ara/rarsM0)
P((,TMr3 +
+
k12Prlrrar,Afo
for 0 < p < 1, 0 < a < 1. Setting p = Erar1/rar1 and a = Erahlrarl'
we have Mrl < EM,, + (k12 T k23) F r,lra rtMO
Hence it suffices to prove that for every integer k z 0 and any a satisfying 0 < at < 1, there is a constant Ck+a depending only on k + a such that (1)
Mk+13 < EMk+x +
Ck+aEk+Q1a/3M0
for 0 < fi < a and 0 < e < 1. Moreover, we claim that (1) follows by induction if there are constants Cx+a such that (2)
Mk+p < EMk+2
i
Ck+aE
fl/00
Mk
0 <#
Mk < EMk+x +
0 <E < 1
0<E<1
Ck+aE11aMk_l
for k 0 and 0 < a < 1. For if (1) holds for all k < j and (2) and (3) hold for all k, then Mi PMt+a + Ci+aP11aMi1 C)vtillMo)
PMs+a + Ci+aP"(aM, +
(where we have employed (1) for k = j  1, fi = 0, and a = 1). Taking p = e/2, a = E11a/2'1a+1C;+a, we obtain (1) for k
fi=0.By(2),if8>0 Mf+r3 < PM,+a
«P t1/a
1_ C'
C
s
rzla QM
+C
vi1%M
by the case of (1) just proved. Taking this time p = E/2 and a = e1( aO)/21+#/(a we obtain (1) for k = j and fi > 0. Since (1) is equivalent to (2) for k = 0, the induction is complete. It therefore remains to prove (2) and (3). If x is any point of T, let x' be another point such that the segment xx' is along the x, axis in W, and set b = Ix  x'I. Then by the theorem of the mean there is a point . such that
IDik10(x)  Dk10(x')1 = Ix  x'I ID (. )I
L. BERS AND M. SCHECHTER
252
Moreover,
Ix  41
D' 0(x)
Hence Dko(x) I < 8"Hx
I
I
< b"H".
2
IID; '
IIco(s')
Since 2R is the diameter of T', we have by summing over i M,t < nR" 6"Mk+" ± 4nR 61M.1 Since e < 1, we may pick x' such that b"n = ER". This gives (3). In order to prove (2) we let x and x' be any two points of V'. Then by definition,
 D'O(x')I
D'O(x)
Ix x'10
< Ix 
x'I"iH",,i[Dko]
where Dk is any derivative of order k. This implies 'P < e. Otherwise RO " Ix  x' I ID ko (X) Rk+,6
(2) if
 Dko(x') I < 2M ak
Ix  x'I1
which gives (2) for this case as well. Thus the proof of Lemma 1 is complete. Proof of Lemma 3. By the nature of the hypothesis and inequality (20) of Chapter 5 we see that we may assume that the diameter of ig is one. Moreover, by Lemma 1, it suffices to prove 1 IID"0IIL,(9)
II0II 'toy) +
CE;'kj
II0IIL,(f)
IPI =i
We first consider a Ca, function f (t) of one real variable having compact support. By the theorem of the mean, we have for any
tand h f "(P) _ for some
f(t + h) f(t) h
such that t < E < t + h. Hence fr IZ,
f'(t)I <
If(T) I dT ±
h
(If(t + h) f(t) 1)
POTENTIAL THEORETICAL APPROACH
253
and z+h
If"(t) Ir <
2r1
(hrif
f"() d+
I f(t + h) f(t) I r
and consequently (4)
f
I f"(T)
,O
if 1(t) r dt < 2rlhr J
x
I r dT + 221hrJ
If(t) Ir dt
Now returning to nspace, let W(x) be a C. function with compact support. We let t be one of the x;, set f (t) = DDW and integrate (4) over the remaining variables. This gives
I
I II DPW II i,() < 2r1hr
IpI=i
II DPW IIi,(I)
1P1=j+1 2r1
hr 1P1=j1
IID"Wlli,(u)
An easy induction argument (as in the proof of Lemma 1) then shows us that there are constants C1 and C2 such that +C2hrj/kj
I IIDD1IIL,(00)
IIWIIL,()
IpI =k
1A =J
If 0 is a Cc0 function in 9, then Ipl=i
IID"
U
.,(91)
< I IIDne0IILr(c0) 1A =j
< C1hr I IID"e0IIL,(c0) +
C2hJ1ki
IIe0IILr(c0)
1A =k
< const. hr
const.
Taking s = const. h given the required result.
hrilki
Il4lli,(9>
CHAPTER 6
Function Theoretical Approach The intimate connection between elliptic partial differential equations and complex function theory became apparent after it was discovered that all solutions of elliptic partial differential
equations with analytic coefficients are themselves analytic functions. The possibility of continuing solutions of analytic elliptic equations into the complex domain gave rise to many important results (Bergman [2], Vekua [2]). But it turned out that, at least in the case of two independent variables, the deep analogy between analytic functions and solutions of elliptic equations does not depend on the analyticity of the equations considered. In particular, one can associate with a second order linear elliptic equation with nonanalytic coefficients a generalized
theory of analytic functionsthe theory of socalled pseudoanalytic functions (Bers [1,8], Vekua [4])which exhibits almost
all characteristic features of classical function theory. In the theory of pseudoanalytic functions, however, one still has to make some smoothness assumptions on the coefficients, though very mild
ones. It is of interest to note that some functiontheoretical results can be obtained without assuming any smoothness or continuity conditions whatsoever. As a matter of fact, consequent application of functiontheoretical methods to elliptic equations, and especially to nonlinear equations, leads one, almost against
one's will, to study linear equations with bounded measurable coefficients (Morrey [2], Bers and Nirenberg [1], Boyarskii [1], cf. also the supplement to Chapter IV in CourantHilbert [2] and the references given there).
In this chapter we report on some results obtained in this direction. Applications will be given in the following chapter. There also exist applications of the theory described in this chapter
to complex function theory itself, in particular, to the theory of Riemann surfaces and discontinuous groups. 254
FUNCTION THEORETICAL APPROACH
6.1.
255
Complex Notation
We shall be concerned with linear elliptic systems of two first order equations for the unknown functions u(x,y), v(x,y) : ux = a11vx + a12vY + b11u + b12v + C1 (1)
uv = a21vx + a22vv + b21u + b22v + C2
and with second order linear elliptic equations for one unknown function O(x,y) : (2)
A110xx + 2A120xv + A22/v1, + A1cx + A20v + A0 6 = B
System (1) is called elliptic if a12 > 0 and (3)
4a12a21
 (all +
a22)2 > 0
uniformly elliptic if all coefficients are uniformly bounded and the lefthand side of (3) is bounded away from zero. If equation (2) is elliptic, we may assume, without loss of generality, that (4)
A11A22A121,
All>0
In this case the equation is called uniformly elliptic if all coefficients are uniformly bounded. We shall always assume uniform ellipticity, but the coefficients a11(x,y), ... , B(x,y) of (1) and of (2) will be assumed only to be
measurable functions. In this case we have no right to expect solutions with continuous derivatives. By a solution (u,v) of (1)
we shall, rather, mean a pair of continuous functions which have L2 derivatives which satisfy the equations almost everywhere. In the case of equation (2) a function 0 will be called a solution if it has continuous derivatives 0x, 0v which in turn have L2 derivatives 0., 0xv, 0vv satisfying the equation almost everywhere. There exists an intimate connection between elliptic equations of the special form (5)
A11Oxx + 2A124xv + A220vv + A14x + A20v = B
and elliptic systems. To see this, associate with every realvalued function 0 its complex gradient w= u + iv = 0x  ic6v. It is clear
256
L. BERS AND M. SCHECHTER
that 0 will be a solution of (5) if and only if (u,v) is a solution of the elliptic system (6)
uz = (2A12/"A11)v + (`422/A11)v  (A1/A11)u + (A2/A11)v ± (B/A11),
uY = vX
Furthermore, system (6) is uniformly elliptic if and only if equation (5) is. The more general equation (2) can be reduced to the special form (5) by introducing the new unknown function c = 0/00, 00 being some fixed positive solution of (2) (B = 0). Under the present weak hypotheses on the coefficients it is by no means obvious that such positive solutions exist. Their existence can, however, be established in every sufficiently small domain, and in every domain if A0 < 0. We shall not, however, discuss this here. We shall make use of the following Let there be given a sequence of functions u" > (x, y) defined in a domain 9' and assume that (i) the sequence u(') LEMMA ON CONVERGENCE.
conrtrges to a function u(x,y) uniformly on every compact subset of 9, (ii) the functions u(2) have L2 derivatives, and (iii) the Dirichlet integrals of the functions u(') over any compact subset of r', are uniformly bounded. Then (j) the function u has L2 derivatives, and (jj) there exists a subsequence u('^) such that u/'  u. and u/ b u, weakly in 'r.
This is an immediate consequence of Lemma 7 of Chapter 4.
A convenient technical simplification results from using complex notations. We set x + iy = z, u + iv = w and denote the conjugate of a complex number by a bar. Functions of x and y (whether real or complexvalued) will be written as functions of z
without implying analytic dependence. The formal derivatives with respect to z and to 2 = x  iy are defined by the relations a __1 a a_1 a a 2(ax+Zay az a2 2(axZTay) In most cases the z and 2derivatives will, of course, be understood to be strong derivatives. The usual rules of calculus apply to these
operators. We note, in particular, that the equation w2 = 0 (7)
FUNCTION THEORETICAL APPROACH
257
equivalent to the CauchyRiemann system (ux  Uy = 0, uy + v = 0) and expresses the fact that w is an analytic function is
of the complex variable z.
An elementary though unpleasantly lengthy computation shows that every uniformly elliptic system (1) can be written in the complex form
Wj=µwz
(8)
where µ(z),
,
vtuZ+aw
;
fltF1+y
y(z) are complexvalued measurable functions satisfying inequalities of the form (9)
. . . ,
lµl + Ivl <_ k < 1,
Ixl + IFII <_ k',
Iyl < k"
Conversely, the complex equation (8) satisfying (9) is always equivalent to a real uniformly elliptic system (1). We note that equation (9) is homogeneous (y  0) if and only if (1) is (ci  c2 = 0), that µ = v = 0 if and only if a12 = a21 = 1, ail = a22 = 0, and that a = = 0 if and only if b;;  0. 6.2.
Beltrami Equation
We consider first a special equation of the form (8), the socalled Beltrami equation
wi = (z)w
(10)
where
IuI
(11)
In real terms equation (10) reads (l Oa)
gux = g12v + gi1vy,
guy = g22vx
+ 912VVI
g2 911912 g12
where the (real) coefficients g,, are defined by (12)
gil
j2
+ 2g12 dx dy + &22 dye = Idz + µ dz12
A Beltrami equation (with a smooth µ) has a simple geometric meaning. A topological mapping w = w(z) satisfying (10) is conformal with respect to the metric (12). This means that if two
258
L. BERS AND M. SCHECHTER
curves in the z plane intersect at an angle 0 (measured in the metric (12) ), then their images in the w plane also intersect at the
angle 0 (measured in the ordinary way). A function satisfying some Beltrami equation is called quasiconformal.
Beltrami equations play an important part in differential geometry, complex function theory, and the theory of differential equations. They occur, for instance, in the problem of reducing
an elliptic equation (5) to canonical form A0 + ... , by introducing new independent variables
computes at once that equation
= E(x,y), n = 77(x,y). One
_ + irk must satisfy the Beltrami
with
µ
A22  All  2iA12 A22 + All + 2
We list now some properties of Beltrami equations. They are proved in the Appendix. (a) If w is a solution of (10) and f (w) an analytic function, f (w (z) ) is a solution of (10).
In the "smooth case" (u of class Ca, w continuously differentiable) the proof is trivial :
af(w(z))/az =f'(w(z))w2 =f'(w)µ(z)wz = u(z) af(w(z))/az (p9) If w(z) is a solution of (l0) in a domain T, if w(z) is a topological mapping of 9, and if w (z) is another solution of (10) in 9, then there exists an analytic function f (w) such that W ^(z) = f (w (z)) .
Again the proof is trivial in the smooth case; one computes easily that aiu/aw = 0. (y) Let w(z) be a solution of (10) which maps the unit disc topologically onto itself leaving the origin fixed. Then
(13)
Iw(z1)
 w(z2)I < K 1z1  z211
Iz1  z21 < K Iw(zl)  w(z2)I' where a and K depend only on k.
This basic result is due to Morrey [2], Ahifors [I], and Lavrent'ev
[1]. The best exponent in (13) is a = (1  k)/(l + k). Mori [1] proved that (13) holds with this exponent and K = 16.
FUNCTION THEORETICAL APPROACH
259
(8) For every measurable u satisfying (11) there exists a unique solution w of (10) which maps the closed unit disc conformally onto itself and satisfies the conditions w(0) = 0, w(1) = 1. (E) If a is a measurable complexvaluedfunction such that la(z) I S k' for I zi < 1, then the nonhomogeneous Beltrami equation
wt =µwa+a
(14)
possesses a unique solution in I z I < 1 which is continuous on I z I < 1, real
on the unit circle, and vanishes at z = 1. This solution satisfies a uniform Holder condition depending only on k and k'.
6.3. A Representation Theorem REPRESENTATION THEOREM (Bers and Nirenberg [1]; cff, also Boyarskii [11). Let w (z) be a solution of the uniformly elliptic equation (8) defined in the domain 9, a subdomain of the unit disc I z I < 1. Then w admits the representation
w(z) = e"ZJ [x(z)] + so(z) where s and so are continuous on I zJ < 1, vanish at z = 1, and are real on IzJ = 1, = x(z) is a homeomorphism (topological mapping) of the closed unit disc onto itself subject to the conditions x(0) = 0, x(1) = 1, and f (?) is analytic in the domain x (9), the image of T under X.
Furthermore, the functions s(z), so(z), x(z) and the inverse homeomorphism x1(4) satisfy uniform Holder conditions depending only on the constants in (9).
If y = 0, the theorem holds with so = 0. If ,u = v  0, the theorem
holds with x(z) = z. If a = # = 0, the theorem holds with s  0. u = v  y  0, the functions Also, in any open set in which a s, so, and x are analytic.
The significance of the theorem lies in the fact that it decomposes an arbitrary solution w of (8) into four functions, three of which are perfectly well behaved and can be estimated without
knowing anything about the particular solution, whereas the fourth is analytic. We remark that the unit disc may be replaced by another domain by means of a conformal transformation.
L. BERS AND M. SCHECHTER
260
We prove the theorem first for a uniformly elliptic equation of the form
w2=uwz±aw+y
(15)
By (e) there exists in (z(< 1 a solution s(z) of the equation
sf=usz±a which satisfies a Holder condition depending on k, k', and the boundary conditions
s = Oatz=1
Ims=0on(z( = 1, We note that
(y es( < kl
where k1 depends only on k, k', k" and set W = e8w. A direct computation (which is legitimate, though we work with strong derivatives) shows that this function satisfies in 9 the equation Wf = uWz + ye8
Again by (e) there exists in IzI < I a solution t(z) of the equation If = utz , ye8 which satisfies a Holder condition depending only on k, k1 and the boundary conditions
t=0atz=1
Imt=Oon(z(=1, We note that the function W = W W2
 t satisfies in T the equation
= ,u W.
By (8) and (y) there exists in (z( < 1 a solution equation
= X(z) of the
XI = YX z 1, satisfies the which maps (z( < 1 topologically onto conditions X(0) = 0, X(1) = 1 and satisfies, together with its
inverse function X1, a Holder condition depending only on k. such that By (f3) there exists an analytic function f T3'(z) =.f [XM] Setting s, = te, we obtain the desired representation.
FUNCTION THEORETICAL APPROACH
261
In order to pass from the special equation (15) to the general case we use a device, due to Morrey, which is applicable only because we admit discontinuous coefficients. Let w(z) be a fixed solution of equation (8) defined in 9. We set µ = u + v(wt/wz) at points of 9 at which wz # 0, and ,u = 0 at points of <9 at which W. = 0 as well as at all points not in 9. Similarly we set
a=a+fl(wfw) at points in at which w 0 and & = 0 at all other points. The functions ,fi(z), &(z) are measurable and
IaI < III + ICI On the other hand, the function w(z) satisfies the equation wf = Pwz + aw + y Ii I < IfuI + IvI,
Hence the validity of the representation theorem in the special case implies its validity in the general case. 6.4.
Consequences of the Representation Theorem
We note some qualitative consequences of the representation theorem. THEOREM ON ZEROS OF ELLIPTIC SYSTEMS.
Let w(z) be a solution of
the uniformly elliptic homogeneous equation (16)
wf=uw=+ytZ+MW +fizz
defined in a domain 9' and assume that w does not vanish identically. Then the zeros of w are isolated.
Without loss of generality we assume that 2' is a subdomain of the unit disc. The representation theorem takes Proof.
the form w (z) = e1cZf [X (Z) J
The analytic function f does not vanish identically. Hence its zeros are isolated and so are the zeros of w, since X is a homeomorphism.
L. BERS AND M. SCHECHTER
262
UNIQUE CONTINUATION THEOREM FOR ELLIPTIC SYSTEMS. Two
solutions of the uniformly elliptic equation (8) coincide everywhere if they
coincide on an infinite set of points which has an accumulation point within their common region of regularity. Proof.
The difference of two solutions of (8) satisfies the
homogeneous equation (16).
In view of connection between second order elliptic equations and elliptic systems noted above we also have the UNIQUE CONTINUATION THEOREM FOR UNIFORMLY ELLIPTIC EQUATIONS.
A solution of a second order uniformly elliptic equation is
uniquely determined by its values in any open set.
Another consequence of the representation theorem is the fact that the maximum principle holds for uniformly elliptic equations of the
form (5) with discontinuous coefficients (in this case the proof in Chapter 2 does not apply). For the sake of brevity we consider only the case of equation (5) with B = 0 and prove that a nonconstant solution has no interior maxima. Indeed, assume that such a
solution c attains its maximum at an interior point, say at z = 0.
Then w = z  ion, = 0 at z = 0 and by the theorem on zeros
of elliptic systems w 0 for 0 < Izj < E. Let the argument of w increase by 2nir as z goes once around the circle I zi = h(0 < h < e)
in the counterclockwise direction. Then n is called the index of w
at z = 0. By the representation theorem w(z) =f [x(z)], where x(z) is a homeomorphism and
f (x) = (x  x(o))m[l + a, (X  x(o)) + a2(x  x(o))2 + . ..], m > 0. This shows that n = m > 0. But it is easy to verify by a geometric argument that at a maximum point we must have n = 1. This contradiction proves our assertion. We conclude this section by a simple but useful EXTENSION OF THE MAXIMUM PRINCIPLE. Let 0 be a solution of the uniformly elliptic equation (5a)
Al1jxx + 2A120., + A2201v = 0
Then the derivatives 0Z, 0y are either constant or have no interior maxima.
(The corresponding statement for more than two dimensions is false.)
FUNCTION THEORETICAL APPROACH
263
The proof is trivial if the coefficients are of class C1 and the solution of class C3, since then 0x and 0y themselves satisfy elliptic equations. (To obtain the equation for 0x, for instance, divide (5a) by A22 and differentiate with respect to x.) In the general case the assertion follows from the representation theorem which implies,
in the case considered, that w = ox
 i/ = f(x(z)) where f is
analytic and X (z) a topological mapping.
6.5. Two Boundary Value Problems
We shall now apply the representation theorem to the Dirichlet and Neumann problems. For the sake of simplicity we consider an equation of the form (5) assumed to be uniformly elliptic and take as our domain the unit disc. The Dirichlet problem consists in finding a solution of (5) under the boundary condition
0 = r on Izi = 1 (r given function) Without loss of generality we assume that r = 0 at z = 1. We (17)
also assume that T has a Holder continuous derivative T' = dT f ds on the unit circumference. Hence we may write (17) in the form (17a)
ao
as
_
T' onlzI=1,
0=0atz=1
The Neumann problem we formulate as follows : (18)
a=c+vonlzl=l,
=0atz=1
Here v is a given function, assumed to be uniformly Holder continuous, c a constant to be determined. This formulation of the Neumann problem has the advantage that it leads to an unrestricted existence and uniqueness theorem. Solutions of both problems are required to belong to C1 (1) . Let 0 be solution of the Dirichlet problem (17) (or of the Neumann problem (18)) for equation (5). Set w(z) _ 0 io,,. A PRIORI ESTIMATE.

There exist positive constants a aid M depending only on the uniform ellipticity of (5) and on the Holder condition for T' (for v) such that (19)
IIWDc.cl> < M
L. BERS AND M. SCHECHTER
264
This estimate is stronger than the Schauder estimate of the preceding chapter since only the size but not the continuity properties of the coefficients are involved.
The proof of the estimate is based on the representation theorem and on a known theorem in function theory, Privalofj's theorem (proved in the Appendix), which states that iff (z), IzI < 1 is analytic and Ref (z) is continuous on I z I = 1 and satisfies there a Holder condition with exponent 6 < 1 and constant K, then f (z) satisfies on Izi < 1 a Holder condition with exponent 6 and constant CK, where C depends only on 6. It was noted above that the function w(z) satisfies equation (6),
i.e., an equation of the form (8) where the complexvalued coefficients u(z), . . . , y(z) are measurable functions satisfying the inequalities (9). The constants k, k', k" depend only on the uniform ellipticity of (5). Applying the representation theorem we have w(z) = e81zf[x(z)] + so(z)
(20)
Here C = x(z) is a liomcomorphism of the closed unit disc onto itself which leaves the points 0, 1 fixed; the functions s(z), so(z) are continuous in the closed unit disc, real on the unit circle, and vanish at z = 1, and the function f is regular analytic for Cl < 1 and is continuous on the unit circle. Also, we know bounds and Holder conditions for the functions s(z), so(z), x(z) as well as for the inverse homeomorphism z which depend only on the constants k, k', k". I
Since T'(e'0) is the tangential derivative of a singlevalued
function it has mean value zero and hence vanishes somewhere on the unit circle, so that by hypothesis (21)
1 r' l < 2K1,
Ki depending on the Holder continuity of T'
Also, from (17) we infer, by considering the variation of 0 on a curve of steepest descent, that there exists a point zl in the unit disc such that (22)
l ic(zi) I < K21
K. depending only on the Holder continuity of T'
FUNCTION THEORETICAI. APPROACH
265
The boundary condition (17a) may be written in the form
Re [izw(z)] = r'(z),
(23)
IzI = 1
or, since s and so are real on Iz1 = 1, (24)
Rc {izf[x(z)]} = e812>[T'(z)  so(z) Re (iz)],
We have that x1 (ei°) = e;n(o)
IzI = 1
0<0<2ir
where the function SA(0) is monotonically increasing, satisfies a known Holder condition, and )(27r) = 27r
S2 (0) = 0,
By Privaloff's theorem there exists an analytic function H(t), I < 1, which is continuous on the closed unit disc and satisfies the conditions

0 < 0 < 2,r,
Re H(1) = 0 and for this function we know a bound and a Holder condition. Im H(ei") = SZ(0)
0,
Set
Re and
so[x1( )] _ s[x1(S)] = a(Q, Condition (24) may be rewritten in the form (25)
Re [i e" ehcc> eof0
Re
c
[ix1(4)])
Noting the Holder continuity of T' and (21) we conclude that the righthand side of this equation satisfies a known Holder condition
and is bounded by a known constant. Applying again the Privaloff theorem we obtain a Holder condition for the regular function i exit f (i) on the closed unit disc. A bound on the real part of this function is known. On the other hand, inequality (22) at a point in together with formula (20) give a bound for If I
the unit disc I I < 1. Hence we also know a bound for the This implies that we know a bound and a function iC e"(; f Holder condition for the function f (4) on the unit circle 1 I = 1.
Applying once more Privaloff's theorem and the maximum
L. BERS AND M. SCHECHTER
266
modulus theorem for analytic functions we obtain a bound and a Holder condition for the function f on I I < 1. Returning to the representation formula (20) and noting that
by now we know Holder conditions and bounds for all terms occurring on the righthand side, we obtain the desired inequality (19).
For the Neumann problem the argument would be the same if we could find a bound on I aolanl. If the equation is homogeneous (B = 0), this is easy, since the maximum principle implies that ac/an must vanish at some point on the boundary and we know
the Holder continuity of aO/an. Also, as before, a0/as must vanish at some point of the boundary, so that we again obtain inequality (22), this time for some point on the unit circumference. For B 0 the argument is more complicated and will be omitted. The Dirichlet and Neumann problems formulated above have unique solutions. EXISTENCE AND UNIQUENESS THEOREM.
Sketch of proof. We consider a sequence of equations (26) A(i)' Ali)ox _ A>i'o = B(i) _l._ 2Aci;J 11 xx
12 xv
= AP + 22
HY
y
with very smooth (say C,o) coefficients such that all equations
(26) are uniformly elliptic with the same constants and the coefficients of (26) converge almost everywhere to the corresponding coefficients of equation (5). Let 0(i) be the solution of (26) under the boundary condition (17) (or (18) ). The existence
of P) is classical; it can be established, for instance, by the methods of Chapter 4 or 5. By our estimate there is an a > 0 and an M such that (27)
II¢ici>Ilcl+a < M
and, selecting if need be a subsequence, we may assume that there is a (28)
satisfying the boundary conditions and such that 11 0  P> II c,
0
It is not difficult to verify (either by elementary means or by the representation theorem) that the Dirichlet integrals of the derivatives of 00) are uniformly bounded over every compact subset in the unit disc. Using the lemma on convergence we conclude that satisfies (5).
FUNCTION THEORETICAL APPROACH
267
Uniqueness can be proved either by the maximum principle or by noting that our derivation of the a priori estimate implies that w= 0 if T' or v vanishes identically. Using the maximum principle and other estimates, which we shall not state here, the solvability of the Dirichlet problem can be proved also for arbitrary continuous boundary values. Appendix.
Properties of the Beltrami Equation. Privaloff's Theorem
This appendix contains a proof of Privaloff's theorem and the theory of the Beltrami equation to the extent needed to establish statements a, of Chapter 6. A solution of (10) of that chapter will be called a euconformal function, a homeomorphic solution, or
a ,uconformal mapping. There are two main problems: to find a uconformal mapping of the whole plane onto itself, and of a disc onto itself. In both cases a mapping exists, and with appropriate normalization, it is unique. For a Holder continuous ,u the existence of local euconformal mappings is classical (Korn [1], Lichtenstein [1]; modernized proofs have been given by Bcrs [7] and Chern [1]). The global mapping theorems follow from the local theorem by use of the general uniformization theorem. Vekua [3] and Ahlfors [2] gave direct proofs of the global theorems. For measurable u the mapping theorem is due to Morrey [2] ; earlier, the case of a continuous u had been treated by Lavrent'ev [2]. New proofs of Morrey's result were given by BcrsNirenberg
[1] and Boyarskii [1]. One owes to Boyarskii the important observation that the generalized derivatives of the solution are of class L9 for some p > 2. The proofs given below are taken from AhlforsBers [ 1 ].
For C.0 functions g(z), z = x define the operators P and T by (Pg)(z) =

IT
(Tg)(z) =  I
iy, witli compact support we
f gm ( 1 ,  LI g(z) d$ dl)
Jf (4  z)2
d$ d??
L. BERS AND M. SCHECHTER
268
where = E ; in and integration is taken over the whole plane. We write II, instead of II i,(x for the sake of brevity. II
II
LEMMA 1.
(1) (2)
The following relations hold:
(Pg) Z = 7'g (Pg); = g, IPg(Z1)  Pg(z2)1 < c Ilgll, IZ1 
for 2 < p < o0 for 1 < p < co 22112/n
(3)
11Te11, < C, IIg1I
(4)
1I T9112 = 119112
Relations (1) are easy to verify; (2) is an immediate consequence of the Holder inequality, whereas (3) is a special case of the CalderonZygmund inequality. The identity (4) follows from (1) and integration by parts. In fact, ffi Tg12 dx dy = ff (Pg) z(Pg)Z dx dy
 f f (Pg)zfPgdxdy = f f (Pg)Z(Pg)zdxdy f f lg12dxdy Let Cp denote the smallest constant for which (3) holds. We have LEMMA2.
C.  l asp, 2.
This is a consequence of the Riesz convexity theorem (M. Riesz [3]) and of (4). It follows from Lemma 1 that for g e L9, p > 1, Tg is defined, while Pg is defined if p > 2. Also Pg belongs to the Banach space B. consisting of complex valued functions w(z), Izl < ee, which
vanish at z = 0, satisfy a Holder condition with exponent 1  (2/p), and have generalized derivatives in L,. The norm in Bp is (5)
II(1)IIBD = H12jp,x[w] ± IIwZII, + IIwOZII,
Clearly P is a bounded linear transformation from L,, to Bp, satisfying (13).
Throughout we shall assume that all functions denoted by µ, with or without subscripts, are bounded and measurable and
FUNCTION THEORETICAL APPROACH
269
subject to the inequality lp(z) I < k with fixed k < 1. The exponent p will be a fixed number satisfying
p > 2,
(6)
kCp < 1
By Lemma 2 there are such p, whatever the value of k. If a E Lp, the equation
THEOREM 1.
cot = µ w z
(7)
+a
has a unique solution w"° E B. Proof. To establish the uniqueness we show that a solution of the homogeneous equation
wz = uw
(8)
reduces to zero if w(0) = 0 and wz E L. It follows by (1) that
w = P(wz) + F where F is holomorphic. We obtain
(0 z = T(µwz) + F
(9)
and by (3) IIF'Ilp < (1 + kCp) Ilwzllp
But IIF'II is finite only if F is constant. Hence F = 0 and (9) yields 11(,)Zllp <
kCp
II(ozIID
a contradiction unless w2 = 0. Thus o) = 0. To prove the existence, we solve the equation
q = T(µq) + Ta
(10)
in L. by setting
q= Ta+TyTa+TuT,uTa+... This is possible because Ta E Lp and the series converges since the norm of the transformation Tju is < kCp < 1. We set (11)
w = P(1uq + a)
and verify that w; = q, w, _ pq + a. Hence co is a solution of (7).
L. BERS AND M. SCHECHTER
270
It follows from (10) that 11g11,
< kCn llgll,, + C, llaj,
and thus llg11, < c Ilall,
(12)
where we use the notation c for unspecified constants which may depend on k and p. This shows that w'',° E B and also establishes The mapping or , o,'`,° is a bounded linear transformation LEMMA 3. from L, to Bn with a bound that depends only on k and p. LEMMA 4. If Fen + Fe almost everywhere and Ilan  611',  0, then co"^. co," in BD.
Proof. We set SP)
oil',' and
qn
q=
(011,(Y. From
fn)
pq + an  a = JunQ(Zn) + l n  lu) q + an  a
= lungn
with An = (Fen  µ)q + an  or. 0 since II (un  u) q II ,,  0 by dominated convergence But II An II n 0 and Il an  a 1l D  * 0 by assumption. Hence, by Lemma 3, f2(n) we conclude that S2('b)
in BD.
In addition to the restriction 1 ,al < k < 1 we assume until the proof of Theorem 3 below that all functions denoted by µ vanish outside a fixed set. There exists a unique µconformal function f' which vanishes at the origin and satisfies f2"  1 E L,. It is given by THEOREM 2.
f'(z) = z f cu1"''(z) It is evident that f" is a solution of
(13)
.fi = ,ufZ Since f'` is locally of class LD with p > 2, it is also locally of class L2. (14)
We remark that f' is independent of the choice of p. Indeed, since u has compact support,, f'' is analytic in the neighborhood of oo. Together with f,"  1 E L, this implies f' = 1 + o(1/Iz12), and hence fi'  1 E Lo for 1 < q < p. LEMMA 5.
If u is a C. function, so is f u. Moreover, f'' is a homeo
morphism of the whole plane onto itself and the Jacobian is positive.
FUNCTION THEORETICAL APPROACH
271
Let us write the Beltrami equation (14) as a system of two real equations for u and v (f = u + iv) Proof.
v. = a12ux + a22u. (15)
Uv = allux + a12uv
(cf. (10a) of Chapter 6). Eliminating v by differentiation, we obtain for u the second order elliptic equation (16)
alluxx  2a12uxv + a22uvv + blux + b2uv = 0
where bl = allx  a12v and b2 = a22,  a12x. By the local existence
theorem (Theorem 1 of Chapter 5) there exists, near every point zo, a C2 solution of (16) with prescribed first derivatives at zo. Finding v by means of (15) we see that in the neighborhood of each point zo there exists a C2 uconformal mapping (z) with positive Jacobian. By Theorem I of Chapter 5, (z) is a C. function. A direct computation shows that f'`(z) is a holomorphic function of (z) near zo. On the other hand, f''(z) is holomorphic for large Izi and since I 1  df'' (z) f dzI " is integrable, we see that
f(z) =z+ao+al/z+... near z = oo. The assertion follows from these statements. of C. functions with fixed It is possible to find a sequence f compact support such that y. > u almost everywhere, and, as a 0. It follows by Lemma 4 that consequence, IIu,,  uII,,
f'n P  0 in BD. We employ this approximation to prove
LEMMA 6. f' is always a homeomorphism of the whole plane onto itself.
Let gn M be the inverse of fn (z) by the preceding lemma. One finds that Proof.
with
gn,r = 1'ngn,Z
vn   (fn, zun) fn.i
Iynl = lun) ° gn
where h  g denotes h (g (Q) .
gn
known to exist
L. BERS AND M. SCHECHTER
272
By use of Holder's inequality we obtain
nl p (If2  Ifa.zl2) dx d).
JfIvV ddi1
S f I 'U. I P2 I J ,,ll2 dX d), d.Y).21p
C where
(fftitnID dx dyll
?1P (55 If,, 21
dx
_ + irk. But Ilf».fllp  II (Vi^,/lnllp
C I!'"nlly
and hence Ilvnlip C C Il,unllp
(17)
It is evident that gn = f"^, since gn, from Lemma 1 and Theorem 2 that Z
W'
Ign(t l) 
1 as z  oo. We conclude C llYnllr
2I1
2/p
or, equivalently, that IZ1  Z21 C Ifn(Z1) f.(Z2)I 1
C
Ifn(zl)
fn(zn)I1_21"
On letting n tend to infinity, we see that f(zl) = f" (z2) implies that z1 = z2i and we have therefore proved that f" is onetoone. The fact that f"  I for z * oo is sufficient to show that the mapping is onto. A homeomorphismf is said to be measurable if measurable sets
are mapped onto measurable sets. An equivalent condition is that null sets are mapped onto null sets, or that the set function mes f (e), defined for Borel sets e, is absolutely continuous. The mapping f" and its inverse are measurable. Moreover, 0 almost everywhere, and
LEMMA 7. fZ`
(18)
mesf"(e) =J.1 (If2I2  If_
for any measurable e.
dxdy
FUNCTION THEORETICAL APPROACH Proof.
273
Let e be an open set. Let x denote the characteristic
function of f'(e) and x that off, (e). Then x < lim inf x and consequently mes f1`(e) < lim inf mes f (e)
We have mesfn(e) =fj.(Ifa.Z12  Ifn,Zl2) dx dy (19)
ff
<
I f,, , =12 dx dy
e
and by Holder's inequality 2IP
J'JIfn.z12 dx dy <
(JJILl2 dx dyl
e
(mes e) l''1n
e
Now
(jjfiP dx
dy)u' Ilfn,Z
 1119 +
(mcs e)11n
e
Cc
II Iu,, 11 P
+ (mes e) 11P
and we obtain 4 (mes e)1/1')2 (mes
(c
e)121n
these estimates show that mes f1'(e) tends to 0 with mes e. Since every null set is contained in an open set of
Because II µn II,  II U ll
arbitrarily small measure, we have proved that null sets are mapped on null sets, and thus f1` is measurable.
Precisely the same reasoning can be applied to the inverse function, since (f")1 = lim f"n and Ilv II,, is uniformly bounded. If a is an open set and bounded, it follows from (19) and the preceding inequality that mes p(e)
(If''I2  I f,112) dx dy
e
since f Z fz' and f,j4
f2 in L2(e). For compact sets e the reverse inequality is true, either by the same argument or by taking complements with respect to a large disc. On approximating
an open set by compact subsets it is easily concluded that the equality (18) holds for all open sets. It will therefore hold for closed sets, for Borel sets, and finally for all measurable sets.
L. BERS AND M. SCHECHTER
274
Because the inverse function maps null sets on null sets, it is
I,ff I2 to vanish on a set of positive measure. It is important to formulate conditions under which a com
impossible for I fi112
posite function h ° fµ = h(f1'(z)) has generalized derivatives. For convenience we simplify the notation p to f. LEMMA 8.
Suppose that hg and ht are locally of class LQ, q Z 2.
Then h of has generalized derivatives given by
(h °.f) t
=
(h x
°f ).f= + (ht °.f )JJ
(20)
(h °.f)2 = (h. °f).fi + (h2 °f)];
and
II(h °f)zIlr S M(IIhzII, + 11h2IIQ),
(21)
r=
pq
P+q2
where the norms are over corresponding bounded regions c, f(S2o), and M is independent of h. Proof.
We show first that (20) implies (21). By Holder's
inequality ffwz
°.f)fIrdxdy
00 <_
(ffvzz° ,fI°
If'12dxdy) (ffv
dxdy)l.1q
and by use of Lemma 7,
°fI°IfI2dxdy <1
1k2
ffJh of jq
(fI2
 1h12) dx dy
fl, 1
k2
Ih214dxdy
A similar estimate applies to (h2 ° f) fZ and (21) follows. The formulas (20) hold if h and f are of class C. By wellknown properties of generalized derivatives (cf. Appendix II to Chapter
FUNCTION THEORETICAL APPROACH
275
1) it is possible to find C. functions hm, fn in such a way that ff(Ihm I(no)
ff(if
 hI2 + Ihm,z  hz12 + Ihm,3  hilt) dx dy . 0 fi
fI2 + Ifn _j2) dxdy
+ ifn.z
0
no
Consider
I, = Jf1(hm.z °f)
 (hz f )I IfsI dx dy
no
12 = f f I (hm.z °f)  (hri.z °fn) I IffI dx dy no
I. = f f Ihm,z
°fnI Ifz fn,ZI dx dy
no
By the same estimate that was used to prove (21) it is seen that I1 can be made arbitrarily small by choosing m sufficiently large. When m is fixed it is evident that I2 and I$ tend to zero for n  oo. Consequently it is possible to choose m and n so large that JJI(hm.z °fn)fn,z
 (hz °f )ffI
dx dy
no
is arbitrarily small. The same'` is true of ffI(hm.2 °fn)Jn,z  (h2 °f)JJI dx dy no
and we conclude easily that the first relation (20) is valid. The second is proved in the same way.
The preceding lemma is used to prove LEMMA 9.
(22)
(fµ)1 = f' with
v =  (L2) ° (f14) i 411
We remark first that v is measurable because f ` is a measurable mapping. Therefore f exists and f' is locally of Proof.
class L..
L. BERS AND M. SCHECHTER
276
We compute (f' o f")t by the second formula of (20) and find that it is zero almost everywhere. Hence ( =f" o f" is analytic, and since it is a homeomorphism, it must be a linear function.
But D(0) = 0 and V(z)  1 as z , oo. Hence (D(z) = z and f' is the inverse off'`. We collect in a single theorem a number of results which are easy consequences of Lemma 8. In this theorem we do not need to assume that y has compact support. Let f be any ,uconformal homeomorphism, defined in a region S2o. The following are true : THEOREM 3.
(i) (ii)
ff is locally of class L.
ff
71 0 almost everywhere.
(iii) f1 has generalized derivatives which are locally of class L. (iv) (f 1)s and (f% are determined by classical formulas. (v) f and f1 transform measurable sets into measurable sets. (vi) Integrals are transformed according to classical rules. (vii) If is any ,uconformal function in 00, then , o f 1 is analytic, and vice versa.
Proof.
Since all assertions are of a local nature, it is no re
striction to assume that SZo is a bounded region. We extend u to the whole plane by setting u = 0 outside S20. The results (ivi) follow from the fact that f o (f') 1 is analytic, and (vii) follows because 0 o (f) 1 and f o (f'') 1 are both analytic.
Statements (x) and (9) of Chapter 6 can now be proved by direct computation, just as in the "smooth" case. We now let w" denote a µconformal homeomorphism of the
whole plane onto itself which is normalized by w"(0) = 0, w''(1) = 1, w''(oc) = oo. Similarly, W" is to be a µconformal homeomorphism of the closed unit disc onto itself which satisfies W"(0) = 0, W''(1) = 1. If they exist, it follows from Theorem 3 that w" and W" are uniquely determined. The existence is proved in several steps. LEMMA 10.
(23)
If u = 0 in a neighborhood of oe, then
w"(z) =f'(z)/f'(1)
If µ = 0 in a neighborhood of the origin, then (24)
w"(z) = 1/w'(1/z)
FUNCTION THEORETICAL APPROACH
277
where
A(z) = 1u(1/z)z2/22 LEMMA 11.
Set u = 91 + µ2 where µl and
1u2
vanish near 0 and
oo, respectively. Then w" = wA O w"=
with
A = (1 `1
(25)
1u21u Ji LEMMA 12.
o (w'')
.f", )
If ,u (z) = µ (1 /2) z2/ z2, then W" is the restriction of
w" to the unit disc.
Lemmas 10 and 11 are proved by direct verification. Together they show that w" exists, for any arbitrary IA can be decomposed in the supposed manner. As for Lemma 12, one proves by means of the uniqueness that w"(z) = 1/V'(1/2). Hence Iw'`(z)I = 1 for
IzI = 1, and it follows that the restriction of w" maps the unit disc onto itself. It is evident that Lemma 12 proves the existence of W". There exist unique ,uconformal homeomorphisms of the whole plane and the unit disc onto themselves with fixed points at 0, 1, o0 and 0, 1, respectively. THEOREM 4.
Theorem 4 contains statement (b) of Chapter 6. LEMMA 13.
If µ = 0 for IzI >_ 1, then c1
Proof.
Isc
<
We have 11p II, < c, and therefore, by (13) and Lemma
3, IzI12/p I,f"(z)I < IzI + c1 By virtue of Lemma 9 and the inequality (17), the same reasoning can be applied to the inverse function and yields
IzI < If'A(z)I These estimates imply
(1 +
+ c1 I f"(z)I12/p
C1)2p2/p2 < Iw"(ej9)I
which proves the lemma.
(1 +
cl)2p2/p2
L. BERS AND M. SCHECHTER
278
Let [zl,z2] denote spherical distance. The following spherical Holder condition holds for all µ. LEMMA 14.
(26)
[w"(zl), w"(z2)] S c[z1,z2]z,
a>0
We represent w" as in Lemma 11, choosing µ1(z) = 0 for Izl > 1, µ1(z) = µ(z) for IzI < 1. Then 121 < kl(k) < 1 and by Lemma 13 together with (25), 2 vanishes in the disc I zl < 1 /c. It is sufficient to prove (26) for w"' and w2. Also, because the spherical distance is invariant under inversion, we may replace wz by w'(z) = 1/wA(1/z), where v(z) = A(1/z)z2/Z2 (see Lemma 10). Thus we need to prove (26) only for the case that µ = 0 for IzI > R; c and a may depend on R. Proof.
We know that If(l) I > c1 (see the proof of Lemma 13). It follows by (23, 13) and Lemma 3 that Iw"(z1)  w"(z2)I < c 1z1 
22112/p
This implies (26) if, for instance, Iz11, Iz21 <_ 3R.
The function g(z) = l/w"(1/z) is holomorphic for IzI < 1/R and if R > 1, as we may suppose, Lemma 13 yields Ig(z) I < c for IzI < 1/R. Hence Ig'(z) I < 4cR for IzI S 1/2R. This implies (26) if I z11, I z21 ? 3R.
Finally, (26) is trivially fulfilled if Iz1I < 2R and Iz21 z 3R. COROLLARY.
I W"(Zl)  W"(Z2) I< C IZ1  Z21"
This implies statement (y) of Chapter 6. We now prove statement (E). The uniqueness part follows at once from Theorem 3. To establish existence, we define u and a to be zero outside the unit disc. Then 11a II
w in the form
7r1l'k'. We represent
w=0+w"'"
We must determine 0 in such a way that Im
on IzI = 1 and
 ('01"') = 0
 co"," = 0 at z = 1. Set /(z) = (D(W"(z)).
We must determine a holomorphic function V(z) in the unit disc such that Im tD = r = Im co"" ° (W")1 on IzI = 1 and D = w"0L
FUNC'T'ION THEORETICAL APPROACH
279
at z = 1. Note that we know Holder conditions for W'`, (W,,)1, w'`.", and r depending only on k and k', as well as a bound on Iw'`"(1) I. It follows from Privaloff's theorem (cf. below) that a 0 with the desired properties exists and satisfies a Holder condition depending only on k and k'. This proves (e). It remains to prove Privaloff's theorem. THEOREM 5 (Privaloff [1]). Let f (z) = u + iv be analytic in 1 and satisfies z I < 1. If u (z) is continuous in I z I u(ezas) I (27) Iu(eia,) < K Ie'e, e'0,I2


then f (z) is continuous in I z I < 1 and satisfies
If(zl) f(z2)I
(28)
where the constant C depends only on x.
Proof.
It clearly suffices to prove that (28) holds for all zi and
z2 such that
Iz21 < 1. For r < 1, we have by Cauchy's
I z1$,
formula,
f(z) =
_rf(')
27r1 J
1
1
d
r2 Z
when IzI < r. First taking the minus sign, we get, after simplification, loan
1
(29)
f (rete) Ireio
=
.f (z)
I ZI12 d8
The plus sign yields 1
(30)
.f (z) = 2ir
j
2a
f(re") 1 + 2ir Im L
= rf (0) +
12rr
IreiO

ze_i
A
z$2
i8 ff(rei0) 2ir Im ze d© Ireie  z12 J
by the mean value theorem. Taking real parts in (29) and imaginary parts in (30), we obtain 2n
u(z) =
2
Io
u(feie)
v(z) = rv(0) +
2
ire
ie
1 2112 de
2 I
i
L. BERS AND M. SCHECHTER
280
or, combining into one formula,
r
f (z) = ira(0) ±
rei°
!
z dO
Since u(z) is continuous in Izl <_ 1, we may let r  1 obtaining .f (z) == iv(0) + 2r
(31)
z
u(es°) eye
j
dO
The proofis now similar to that of the HolderKornLichtenstein
Giraud inequality (see Appendix I to Chapter 5). It clearly suffices to prove (28) for 1 z11, I Z21 > 1 /2 and 6 = I zl  z21 < 1/3.
Let Al be the (possibly empty) set of those 0 in [c1  ir, i ; 7r] satisfying le'0  z,l < 26, and let 02 be its complement in that interval. For I z1 < 1 2 e'0 + z
1
2Tr
o
ei0
z d0=1
Hence if zl = rle'A' and zz 
then
f (zi) f(z2) = u(e''')  u(e'k1)
f 1
d1+n([u(e'o)
(32)
1

u(e'41)]
+ [u(e'0)  u(e'o')] e,o

e
Since I z, 1, I Z21 > 1/2, le"01

e'411
e  zl
Z2} dO z2
< ?t I Z2  zl I
Hence by (27) l u(e'0=)  u(e'41) I < K20'6"
(33)
We express the integral in (32) as the sum of two integrals Il and
I2 with integration taken over Al and :,2, respectively. Since 6 < 5, there is a constant c such that le;°

etb;l < 10
 oil < c Iei°  z,I,
j = 1, 2
FUNCTION THEORETICAL APPROACH
for ci  i < b <
; ir. Hence by (27) K
1111 <
281
77C n
f
,1
(10

10
 0211)
dO
(34)
const. b"
Moreover,
12 =
(zl

[u(e'B)
z2)

to (a to
 z1) (e1  42)
Now for 0 E7r A21 le=e
 z1 < 2 Ie'e  z21
Hence 1121
10 7TK6 C
4 I I"2 de
S const. ax
The result now follows immediately from (33), (34), and (35).
CHAPTER 7
QuasiLinear Equations In discussing nonlinear elliptic equations we shall limit ourselves to second order quasilinear equations of the form
+ 2B(... )0xv + C(... )#vy = D(...
(1)
A, B, C, D being continuous functions of their five variables which satisfy the ellipticity condition
ACB2>0
(2)
in the domain considered. Typical examples are the equation AO
(3)
= D(x,y,0,0¢,Ov)
the equation of minimal surfaces (4)
(1 + O2%.  20,OyO
v
+ (1 + Ox)Oyv = 0
and the equation of a potential gas flow (5)
(P&)x + (Pov)v = 0,
p=1y
2
(02,
1
+
Oy)I'v1
where y > 1 is a constant. The latter equation is elliptic only if the flow is subsonic, i.e., if Ol + Oy < 2/(y 1).

7.1.
Boundary Value Problems
In studying boundary value problems for equation (1) we shall assume, merely for the sake of simplicity, that the domain considered is the unit disc I z I < 1 (x + iy = z). We shall consider the Dirichlet problem
(6a)
= r (given function) on 1': 282
IzI = 1
QUASILINEAR EQUATIONS
283
and the Neumann problem which is convenient to formulate as follows :
ao = c + y (constant + given function) on I',
(6b)
0 = 0 at 1 (The constant c is not prescribed but is to be determined.) In order to establish uniqueness theorems for a boundary value
problem one usually makes use of the obvious but important observation that if the coefficients of (1) are continuously differen
tiable, the difference co = 01  02 of two solutions satisfies a linear elliptic equation. In fact, we have that i = 1, 2
A1o:,xx + 2B,0t>xv + CA,vv + D, = 0,
where A; = A(xy,0t,0i,x,0t,,,), etc. Subtracting the two equations, we obtain that (7)
A2(k)x, + 2B2wxv + C2wvv + (A2
 A1) 01,
+ 2(B2  B1)01,xv + (C2  C1)01,vv + (D2  D1) = 0. The differences (A2  A1), ... , can be computed as follows. Set
F(t) = A[x,y, 02 + (1  t)01, 02,x + (1  t)41,x, 02,v + (1  t)01,vl Then
dt + to,, f
A2  Al = fol F' (t) dt = co f lA,6 dt + (1,S f
dt o1A,.
o
Thus (7) may be written in the form (8)
A2w= + 2B2wxv + C2wyy + awt + bw + cw = 0
where 1
a = 01, faAd= dt + 201, fo
1
1
dt + 01.vv oCo= di ±
1
o
D,= dt
and b and c are defined similarly. Uniqueness of the Dirichlet or Neumann problem will be established if we can assert that the
284
L. BERS AND M. SCHECHTER
linear equation (8) has only the trivial solution u.) = 0 under the homogeneous boundary conditions. This is so, for instance, if A, B, C do not depend on 0 and aD/ao S 0. In this case c < 0 and the maximum principle is applicable (cf. Chapter 2).
7.2.
Methods of Solution
Existence proofs for our two boundary value problems can be reduced to the search of a fixed point of a transformation in an appropriately chosen Banach space B, the elements of which are functions defined in the unit disc. Let 'D(x,y) be an element of B. We form the linear equation (9)
airy + 2box, ± co,, = d
where
a(x,y) = A(x,y,t,(x,(D), .. .
and solve for it the boundary value problem considered. (We assume, of course, that this is possible, and in a unique way.) Denote the solution by ¢ = TO and assume that it belongs to B. Then T is a (nonlinear) mapping of B into itself and a fixed point
of T, i.e., a 0 such that 0 = To is a solution of the nonlinear boundary value problem. We will now describe several methods for proving the existence of a fixed point of T or, what is the same, of a root of the equation (10)
K(D=(1T)(D =0
(a) Successive Approximations.
One may try to find a fixed point
of T by choosing some elements 0 and forming successively 01 = T O O , D2 = T01i etc. If the method converges, that is if (D)
'V and if T is continuous, then TO = (D. The convergence of the method of successive approximations
is often assured by the socalled principle of contracting mappings :
If T maps a closed convex subsets of a Banach space B into itself, and if there exists a constant 0, 0 < 0 < I such that (11)
IIT'V1  T'V211 <_ 0 110 1 'V211
QUASILINEAR EQUATIONS
285
then T has a unique fixed point which can be found by the method of successive approximations. There is hardly any need to write down the perfectly obvious proof. We have already applied this method several times in Chapter
5. There we dealt with linear operators T and condition (11) took on the form IITII < 1. Application of the method of succes
sive approximations to nonlinear equations of the type of (2) are classical and will be found in several texts. In this method one embeds equation (10) into a oneparameter family of equations (b) Continuity Method.
(12)
KjO=0,
0
in such a way that K depends continuously on t, Ki = K and the equation KO O = 0 is easily solvable. Denote the set of t for which equation (12) is solvable by r; r contains 0 and one wants to show that T is both open and closed.
In order to show that r is closed one needs to find a priori estimates of the solutions of (12) which imply that all solutions lie in a compact subset Zo of B. Then if t, c T and t,  to one can conclude that to c r. In fact, there will exist elements , in lo with Kt,(D, = 0. Since Eo is compact we may assume, selecting
if need be a subsequence, that 0, , $o. Then K0O0 = 0, so that to E r.
In order to show that r is open one usually tries to show that if a zero of K,0 is known, a zero of Klo +, can be found by the method of successive approximations for e sufficiently small.
We have given an example of the continuity method, for a linear problem, at the end of Chapter 5. (c) LeraySchauder [1] Method. This is a more sophisticated version of the continuity method. We assume now that T has the
property of mapping every bounded set into a compact set (complete continuity). Under this hypotheses, if T is defined on the
closure r of a bounded domain and K
0 on the boundary
E of E, Leray and Schauder define an integer d = d(E,K) which measures, intuitively speaking, the algebraic number of zeros of K in Z. The "degree" d is 0 if K 0 0 in Z. If K maps Z topo
logically onto a domain containing the origin, then d = *1. The most important property of the degree is its homotopy
286
L. BERS AND M. SCHECHTER
invariance: d does not change if K is changed continuously, provided that no zero of K appear on E during the transformation. Hence we can prove the existence of a solution if we succeed to deform T continuously into a mapping To in such a way that no zero of K appear on the boundary during the deformation and the degree of KO = 1  To is 0. For quasilinear equations it is often sufficient to use a somewhat simpler method described below. (d) The BirkhofKellogg [ 1 ]Schauder [3] Method is based on the extension to infinitely dimensional spaces of the wellknown fixed
point theorem of Brouwer. This extension (Schauder's fixed point theorem) reads: If T is a continuous mapping of a closed convex set Z in a Banach space B into a compact subset Eo E, then T has a fixed point in E.
Note that for the particular mapping T with which we are concerned we ought to have no difficulty in establishing complete
continuity. In fact, 0 = TO is a solution of a linear elliptic equation the coefficients of which depend on (D and we know that a solution is "smoother" than the coefficients. The main difficulty lies in finding sufficiently strong a priori estimates which permit
one to apply the LeraySchauder method, or the Schauder fixed point theorem. In general, the Schauder fixed point theorem can be used if the a priori estimates come from the theory of linear equations; if the estimates are derived from properties of nonlinear equations, the LeraySchauder method must be employed. 7.3. Examples
We shall now apply method (d) to the Dirichlet and Neumann problems. We consider first an equation of the form (13)
+ 2B( ... %1, + C( ... )0,, = 0
and assume that the equation is uniformly elliptic (that means that
the coefficients are bounded and AC  B2 z const. > 0). We also assume that the boundary functions T and v in (6) are of class Cl_, and C2, respectively. Finally, and for the sake of brevity,
we assume that A, B, C are Holder continuous in their five
QUASILINEAR EQUATIONS
287
variables, though mere continuity would suffice (cf. BersNirenberg [2]. For an application of the same idea to gas dynamics see Bers [6]).
For B we take the space Ci+a (1), fi to be determined later. If 0 e C,.,_,,(1) and we form the linear equation
A(x,y,O,(Dx,O.)0zz + 2B(... ) 0., + C( ... )0y, = 0 then, according to the results of Chapter 6, our Dirichlet problem (or our Neumann problem) will have a unique solution 0 = TO. If we choose y sufficiently small we will have (14)
(15)
II0 II c,+Y
=
11TO ll el+V
<M
where M depends only on the uniform ellipticity of (13), the boundary function, and y but not on 0. Since the coefficients of (14) are Holder continuous, 0 will have Holder continuous second derivatives.
We choose
j9
so that 0 < fI < y.
Since
114IIc1,R <_ const. Ilollc,+,,
we see that T maps the closed convex set E :
II
Il
c,+ < Ml
of C,+,, (1) into itself and more precisely into the convex subset Ea
characterized by (15). But this subset is compact in Cl_, by Lemma 2 of Chapter 5. Next, T is continuous. For suppose that we have a sequence 0, in E and that 110, Oil c,+ ft , 0. Then there is a subsequence oo, ;n = Tl;y {.;n} such that for n in the Cl+. sense. We note that 0 satisfies the required boundary conditions. Also, the coefficients of the (linear) equation satisfied by 0;. converge to the coefficients of (14) and using the estimate in Theorem 3 of

Chapter 5 we can conclude that 0 satisfies (14). Thus 0 = TO and since TO is uniquely determined the selection of a subsequence
was unnecessary and we have that TED,  TO. Thus all hypotheses of the Schauder theorem are satisfied and we conclude that the Dirichlet and Neumann problems have solutions.
Note that we established existence under weaker conditions than those needed for a uniqueness proof. The argument would be almost the same if we were to assume
that the coefficients of (13) are only continuous. In this case
288
L. BERS AND M. SCHECHTER
solutions of (14) would have generalized L2 derivatives of second
order and the lemma on convergence from Chapter 6 would have to be used in establishing the continuity of T.
The same method works for equations of the form (1) if we require, for instance, that the righthand side D( ... ) should satisfy the inequality K(1 + 10.1 +
(16)
For in this case every solution of (9) may be considered as a solution of a uniformly elliptic linear equation of the form (17)
acx,
 2boxIl + co., + dcz + ecY = f
and we may again apply the estimates of Chapter 5.
We note also that in the case of the Dirichlet problem the assumption that the boundary function be Holder continuously
differentiable can be removed by approximating the given continuous boundary data by functions of class C1i
The assumption of uniform ellipticity, which is not satisfied
by such important equations as the equation (5) of minimal surfaces, can be removed in the case of the Dirichlet problem if we restrict ourselves (1) to equations of the form (13), (2) to boundary data of class C.2, and (3) to strictly convex domains bounded by a curve having continuous curvature. (In the previous results we worked with the unit disc for convenience only. In the case of the Dirichlet problem our argument could be carried out even for multiply connected domains.) We consider again the Dirichlet problem (6a) for equation (13) without assuming uniform ellipticity. Suppose we knew that for a fixed r of class C2 the function 0 = TO (the solution of (14)
satisfying (6a)) satisfies the inequality (18)
II0IIc, <_ A1o
A10 being independent of 1. It is easy to construct a uniformly elliptic equation `1(X3',0,0s,Oti)Orz i 2B( ... )dry + C( ... )OYY = 0 such that (19) is identical with (13) for 11011r, < .110. This new equation has, by our previous result, a solution 0 satisfying (6a). This function is, however, also a solution of (13). (19)
QUASILINEAR EQUATIONS
289
Thus everything hinges on establishing the following A PRIORI ESTIMATE. Let 0 be of class C, on x2 ; y2 < 1, of class C2 on x2 y2 < 1, and let it satisfy an elliptic (not necessarily uniformly elliptic) equation (20)
aox= + 2b¢xv + cOv,
=0
Furthermore, assume that 0= Ton x2 + y2 = 1, T being of class C2. Then (18) holds, M0 depending only on T.
This is often proved by using the socalled Radov. Neumann threepointtheorem. A simpler argument, going back probably to S. Bernstein, is as follows. First of all, 141 cmaxITI by the maximum principle. Since T is twice continuously differentiable we can find for every point (xo,yo) on the unit circle six numbers xi, fl1, YD 72, F'2, y2 such that (21)
gq
(22)
7i2
1
,
2
2
2
#i, 7? T 72
Kg
(K0 a constant independent of x0i (23)
alxo + P1yo + yl = 72x0 + P2yo + Y2 = T (xo)y0)
and (24)
7,x + I'ly F Y1 < T (X,_)') < a2x + 192J + Y2 for x2
ry = 1 2
[Geometrically speaking: the planes 0 = 7,x + /3,y i y, have uniformly bounded slope, pass through the point (x0,yo) and "enclose" the boundary curve of the surface 0 = O(x,)).] Since every linear function satisfies (20), the maximum principle and (24) imply that (25)
71x + fl1y + Y1 < 0(x,y) < x2x + /2}' J 72
so that we obtain from (23) and (22) the inequality I
anI SK0
290
L. BERS AND M. SCHECHTER
for the normal derivative. The tangential derivative of 0 is bounded by that of T. Hence there is a constant Kl such that
on r
10.1 + I0yI < Ki
Since 0z and 0, satisfy the maximum principle (cf. Chapter 6) (26)
10.1 + Ioyl s Ki
everywhere, and from (21), (24), we obtain (18). In the present circumstances (i.e., without uniform ellipticity)
the condition that r be smooth cannot be removed, as can be shown by an example (Finn [1,2]). We have given only a few examples of nonlinear existence
theorems. Many more will be found in the literaturebut it must be kept in mind that all such proofs (including those obtained by variational methods which we had no occasion to discuss) are
based on a priori estimates. It was the absence of methods for obtaining sufficiently strong a priori estimates in more than two dimensions which up to now restricted the theory of nonlinear elliptic equations to the plane. Quite recently, however, something
of a breakthrough has been achieved. On the one hand, one possesses now a priori estimates and existence proofs for quasilinear
second order equations which have coefficients deviating not too
much from constants (Cordes [2], Nirenberg, and, for the gas dynamical equation, also Finn and Gilbarg [2]). On the other hand, De Giorgi [1] and Nash [1] obtained independently an interior a priori estimate for the Holder continuity of the solution of a uniformly elliptic equation written in divergence form
=0 the estimate depending only on max 101 (or the L2 norm of 0) and the ellipticity of the equation (cf. also Morrey [3], Stampacchia [2]). This work has been simplified and extended by Moser [1]. We have, therefore, good reasons to believe that significant progress in the general theory of quasilinear elliptic equations is due in the near future.
291
Bibliography Agmon, Shmuel
1. Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc., 66, 7780 (1960).
2. The L. approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa (3) 13, 4992 (1959). 3. The coerciveness problem for integrodifferential forms, J. Analyse Math., 6, 183223 (1958).
4. Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane I, Comm. Pure Appl. Math., 10, 179239 (1957).
Agmon, Shmuel, Douglis, Avron, and Nirenberg, Louis 1. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12, 623727 (1959). Ahlfors, Lars 1. On quasiconformal mappings, J. Analyse Math., 4, 158 (1954). 2. Conformality with respect to Riemannian matrices, Ann. Acad. Sci. Fenn Ser. Al, 206, 122 (1955). Ahlfors, Lars and Bers, Lipman
1. Riemann's mapping theorem for variable metrics, Ann. Math., 72, 385 404 (1960).
Aronszajn, Nachman 1. A unique continuation theorem for solutions of elliptic partial differential
equations or inequalities of second order, J. Math. Pures Appl. (9) 36, 235249 (1957). 2. On coercive integrodifferential quadratic forms, Conference on Partial Differential Equations, Technical Report No. 14, University of Kansas, 1954, pp. 94106. Berg, P. W. 1. On the existence of Helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15, 289347 (1962). Bergman, Stefan
1. Jber die Entwicklung der harmonischen Functionen der Ebene and des Raumes nach Orthogonalfunctionen. Math. Ann., 86, 238271 (1922).
2. Linear operators in the theory of partial differential equations, Trans. Amer. Alath. Soc., 53, 130155 (1943).
Bergman, Stefan and Schiffer, M. M. 1. Kernel functions and elliptic differential equations in mathematical physics, Academic Press, New York, 1953. Berstein, Serge
1. Sur la nature analytique des solutions des equations aux derivees partielles due second ordre, Afath. Ann., 59, 2076 (1904).
L. BERS AND M. SCHECHTER
292
2. Demonstration du theoremc de M. Hilbert sur la nature analytique des solutions des equations du type elliptique sans l'emploi des series normales. Math. Z., 29, 330348 (1928). Bers. Lipman 1. Theory of Pseudoanalytic Functions. Lecture Notes, New York, 1953.
2. On mildly nonlinear partial differential equations of elliptic type, J. Res. Nat. Bur. Standards. 51, 229236 (1953).
3. Univalent solutions of linear equations, Comm. Pure Appl. Afath., 6, 513526 (1953). 4. Function theoretic properties of solutions of partial differential equations of elliptic type, Ann. Math. Studies, No. 33, Princeton, 69 94 (1954). 5. Local behavior of solutions of general elliptic equations, Comm. Pure App!. Math., 8, 473 496 (1955). 6. Existence and uniqueness of a subsonic flow past a given profile, ibid., 7, 441504 (1954). 7. Riemann Surfaces (mimeographed lecture notes), New York University (195738). 8. An outline of the theory of pscudoanalytic functions, Bull. Amer. Math. Soc., 62, 291 331 (1956). Bers, Lipman and Nirenberg, Louis 1. On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, Atti del Conregno internazionale sulle Equazionz alle derirate parzialli, Trieste, 1954, pp. I1 1 140.
2. On linear and nonlinear elliptic boundary value problems in the plane, ibid., pp. 141167. Bitsadzr, A. V. 1. On the uniqueness of solutions of the Dirichlet problem for elliptic partial differential equations, Uspehi Alai. Nauk. 3, 211 212 (1948) (Russian). Birkhoff, G. D. and Kellogg, O. D. 1. Invariant points in function space, Trans. Amer. Math. Soc.. 23, 96115 (1922).
Bochner, Salomon
1. L'ber orthogonale Systcme analytischcn Functionen, Math. Z., 14, 180 207 (19221. Boyarskii. B. V.
1. Generalized solutions of systems of differential equations of first order
and elliptic type with discontinuous coefficients, Afat. Sb.. 43 (85), 451 503 (1957) (Russian). Browder, F. E. 1. Strongly elliptic systems of differential equations, .4nn. Afath. Studies, No. 33. Princeton, 1551 (19541.
2. On regularity properties of solutions of elliptic differential equations, Comm. Pure Appl..11ath. , 9, 351361 (1956).
3. A priori estimates for solutions of elliptic boundary value problems, Nederl. Akad. lVetenrch. Indag..11ath.. 22, 145169 (1960).
4. Estimates and existence theorems for elliptic boundary value problems, Proc. Vat..4cad. Sci., U.S.A., 45, 365 372 (1959).
BIBLIOGRAPHY
293
Calderon, A. P.
1. Uniqueness of the Cauchy problem for partial differential equations, Amer. J. Math., 80, 1636 (1958). Calderon, A. P. and Zygmund, Antoni 1. On the existence of certain singular integrals, Acta Math., 88, 85139 (1952).
2. On singular integrals, Amer. J. Math., 78, 289309 (1956). 3. Singular integral operators and differential equations, ibid., 79, 901921 (1957).
Carleman, Torsten 1. Sur les systemcs lincaires aux d6riv6es partielles du premier ordre a deux variables, C. R. Acad. Sci. Paris, 197, 471474 (1933).
2. Sur un probkme d'unicit6 pour les systc'mes d'equations aux &rivees partielles a deux variables independentes, Ark. Alai., 26B, 19 (19391). Chern, S. S. 1. An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc., 6, 781782 (1955).
Cohen, P. J. 1. The nonuniqueness of the Cauchy problem, Technical Report No. 93, Applied Mathematics and Statistics Laboratory, Stanford University, 1960.
Cordes, H. O. 1. Uber die eindeutige Bestimmtheit der Losungen Elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. 11ss. Gottingen Math.Phys., Kl. IIa, 239258 (1956).
2. Uber die erste Randwertuafgabc bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 131, 278 312 (1956). Courant, Richard 1. Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience,
New York, 1950.
Courant, Richard, Friedrichs, K. 0., and Lewy, Hans 1. Ober die partiellen Differenzgleichungen der mathematischen Physik, Math. Ann., 100, 3274 (1928).
Courant, Richard and Hilbert, David 1. Methoden der :Matematischen Physzk, Springer, Berlin, 1937.
English
translation of Vol. I published by Interscience, New York, 1953. 2. Methods of Mathematical Physics, II, Interscience, New York, 1962. De Giorgi, Ennio 1. Sulla differenziabilita e l'analiticita delle estremali degli intcgrali multipli regolari, Mem. Accad. Sci. Torino, (3a) 3, Parte I. 2543 (1957). Douglis, Avron 1. Uniqueness in Cauchy problems for elliptic systems of equations, Comm. Pure Appl. Math., 6, 291298 (1953). 13, 593607 (1960). Douglis, Avron and Nirenberg, Louis
1. Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8, 503 538 (1955).
L. BERS AND M. SCHECHTER
294
Fichera, Gaetano
1. Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parzialle lineari, Atti del Convego internazionale sulle Equazioni alle derivate parzialli, Trieste, 1954, pp. 174227.
Finn, Robert
1. On equations of minimal surface type, Ann. Math., 60, 397416 (1954).
2. Growth properties of solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 9, 415423 (1956).
Finn, Robert and Gilbarg, David 1. Asymptotic behavior and uniqueness of plain subsonic flows, Comm. Pure Appl. Math., 10, 2363 (1957). 2. Three dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98, 265296 (1957).
Frank, Philipp and v. Mises, Richard 1. Die Differential and Integralgleichungen der Mechanik and Physik, Vol. I, Mary S. Rosenberg, New York, 1943.
Friedrichs, K. O. 1. The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc., 55, 132151 (1944). 2. On the differentiability of solutions of linear elliptic differential equations, Comm. Pure Appl. Math., 6, 299326 (1953).
Garding, Lars
1. Dirichlet's problem for linear elliptic partial differential Equations, Math. Scand., 1, 5572 (1953). Giraud, Georges 1. ProblZmes mixtes et problPmes sur de varietes closes relativement aux equations lineaires du type elliptique, Ann. Soc. Polon. Math., 12, 3554 (1934).
2. Generalizations des problcmes sur les operations du type elliptique, Bull. Math. Soc. France, 56, 248272, 281312, 316352 (1932).
3. Problemes de valeurs a la frontiere relatifs a certaines donnces discontinues, ibid., 61, 154 (1933). Guseva, O. V. 1. On boundary problems for strongly elliptic systems, Dokl. Akad. Nauk SSSR, 102, 10691072 (1955) (Russian).
Hardy, G. H. and Littlewood, J. E. 1. Some properties of fractional integrals, Math. Z., 27, 565606 (1928). Hartman, Phillip and Wintner, Aurel 1. On the local behavior of solutions of nonparabolic partial differential equations, Amer. J. Math., 77, 453483 (1955). Heinz, Erhard 1. fiber die Eindentigkeit beim Cauchyschen Anfangswertproblem eines elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad. Wiss. Gottingen, Math.Phys., Kl. Ha, 112 (1955).
Hopf, Eberhard 1. Ober den funktionalen, insbesondere den analytischen Charakter, der
BIBLIOGRAPHY
295
Losungen elliptischer differentialgleichungen zweiter ordnung, Math. Z., 34, 194233 (1931).
2. Elementare Bemerkungen Tuber die Losungen partiellen Differentialgleichungen zweiter Ordnung vom Elliptischen Typus, Preus, Akad. Wiss. Sitzungsber., 19, 147152 (1927).
Hormander, L. V. 1. On the uniqueness of the Cauchy problem, Math. Scand., 6, 213225 (1958) ; 7, 177190 (1959).
John, Fritz 1. Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955.
2. Derivatives of continuous weak solutions of linear elliptic equations, Comm. Pure Appl. Math., 6, 327335 (1953).
Kellogg, O. D. 1. Foundations of Potential Theory, Springer, Berlin, 1929; Dover, New York, 1953.
Korn, Arthur 1. Zwei Anwendungen der Methode der sukzessiven Annaherungen, Schwarz Festschrift, Berlin, 215229 (1919).
Kondrashov, V. I. 1. Some estimates for families of functions satisfying integral inequalities, Dokl. Akad. Nauk SSSR, 18, 235240 (1938) (Russian). Koselev, A. I. 1. A priori L. estimates and generalized solutions of elliptic equations and systems, U.spehi Mat. Nauk, 13, 2989 (1958) (Russian). Ladyzenskaya, O. A. 1. On the closure of an elliptic operator, Dokl. Akad. Nauk SSSR, 79, 723725 (1951) (Russian). Landis, E. M. 1. On some properties of solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 107, 640643 (1956) (Russian). Lavrent'ev, M. A.
1. A general problem of the theory of quasiconformal representation of plane regions, Mat. Sb., 21, 285320 (1947) (Russian). 2. Sur une classe des representations continues, Mat. Sb., 42, 407434 (1935).
Lax, P. D. 1. On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math., 8,615633 (1955).
2. A stability theorem for solutions of abstract differential equations and its application to the study of local behavior of solutions of elliptic equations, ibid., 9, 747766 (1956). Lax, P. D. and Milgram, A. N.
1. Parabolic equations, Ann. Math. Studies, No. 33, Princteon, 167190 (1954).
Leray, Jean and Schauder, Julius 1. Topologie et equations functionelles, Ann. Sci. Ecole Norm. Sup., 51, 4578 (1934).
L. BERS AND M. SCHECHTER
296
Lewy, Hans
1. 1\euer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen, Math. Ann., 101, 609619 (1929). Lichtenstein, Leon
1. Zur Theorie der Konformen Abbildungen. Konforme Abbildungen nichtanalytischer singularitatenfreirer Flachenstucke auf ebene Gebiete, Bull. Acad. Sci. Cracoi,ie, 192217 (1916).
Lions, J. L.
1. Problemes aux limites en theorie des distributions, Acta Math., 94, 1153 (1955). 2. Lectures on elliptic differential equations, Tata Institute of Fundamental Research, Bombay, 1957.
Lopatinskii, Ya. B. 1. On a method of reducing boundary problems for a system of differential equations of elliptic type to regular equations, Ukrain. Mat. Z., 5, 123151 (1953). Magenes, Enrico
1. Sul teorema dell'alternativa nei problemi misti per le equazioni linear ellittiche del secondo ordine, Ann. Scuola Norm. Sup. Pisa (3) 9, 161200 (1955). Malgrange, Bernard
1. Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution, Ann, Inst. Fourier, Grenoble, 6, 271355 (19556).
Miranda, Carlo 1. Equazioni Delle Derivate Parziali di Tipo Ellitico, Springer, Berlin, 1955.
2. Teorema del massimo modulo e teorema di esistenza e di unicita per it problema di Dirichlet relativo alle equazoni ellittiche in due variabili, Ann. Mat. Pura Appl., (4) 46, 265312 (1958).
3. Sui problemi misti per le equazoni lineari ellittiche, Ann. Mat. Pura Appl., (4) 39, 279303 (1955). Mizohata, Sigeru
1. Unicite du prolongement des solutions des equations elliptiques du quatrieme ordre, Proc. Japan Acad., 34, 687692 (1958). Mori, Akira 1. On an absolute constant in the theory of quasiconformal mappings, J. Math. Soc. Japan, 8, 156166 (1956). Morrey, Jr., C. B. 1. On the analyticity of the solutions of analytic nonlinear elliptic systems of partial differential equations, Amer. J. Math., 80, 198237 (1958). 2. On the solutions of quasilinear elliptic partial differential equations, Trans. Amer. Math. Soc., 43, 126166 (1938). 3. Second order elliptic equations in several variables and Holder continuity, Math. Z., 72, 146164 (1959). Morrey, Jr., C. B. and Nirenberg. Louis
1. On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math., 10, 271290 (1957).
BIBLIOGRAPHY
297
Moser, Jurgen 1. A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure App!. Alath., 13, 457468 (1960).
Motzkin, T. S. and Wasow, Wolfgang 1. On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys., 31, 253259 (1953). :Miller, Claus 1. On the behavior of the solutions of the differential equation Au = F(x,u) in the neighborhood of a point, Comm. Pure Appl. Math., 7, 505551 (1954).
Nash, John 1. Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80, 931954 (1958). von Neumann, John
1. Jber einen Hilfssatz der Variationsrechnung, Abh. Math. Sem. Univ. Hamburg, 8, 2831 (1931) .
Nikodym. O. M. 1. Sur un theoreme de M. Zaremba concernant les fonctions harmoniques, J. Math. Pures App!., 12, 95108 (1933). Nirenberg, Louis 1. Remarks on strongly elliptic partial differential equations, Comm. Pure App!. Math.. 8, 648674 (1955). 2. Uniqueness of Cauchy problems for differential equations with constant leading coefficients. Comm. Pure App!. Math., 10, 89105 (1957).
3. On a generalization of quasiconformal mappings and its application to elliptic partial differential equations, Ann. Math. Studies, No. 33, Princeton, 95100 (1954).
4. Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math., 9, 509530 (1956). Pederson, R. N.
1. On the unique continuation theorem for certain second and fourth order equations, Comm. Pure App!. Math., 11, 6780 (1958).
Peetrc, Jaak 1. `I'heoremes de regularite pour quelques classes d'operateurs differentiels, Thesis, Lund, 1959.
2. Mixed problems for higher order elliptic equations in two variables, I, 11, Ann. Scuola Norm. Sup. Pisa, (3) 15, 337353 (1961) ; (3) 17, 112 (1963). Petrowsky, I. G.
1. Sur l'analyticite des solutions des systemes d'equations differentielles, Afat. Sb. N.S., 5 (47). 370 (1939). du Plessis, Nicolaas 1. Some theorems about the Riesz fractional integral, Trans. Amer. Math. Soc., 8, 124134 (1955). Plis, Andrew
1. Nonuniqueness in Cauchy's problem for differential equations of elliptic type, J. Math. Mech., 9, 557562 (1960). 2. A smooth linear elliptic differential equation without any solution in a sphere, Comm. Pure App!. Math., 14, 599617 (196 1) .
L. BERS AND M. SCHECHTER
298
Privaloff, J. 1. Sur les fonctions conjugees, Bull. Soc. Math. France, 44, 100103 (1916).
Protter, M. H. 1. Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95, 8191 (1960). Rado, Tibor 1. Geometrische Betrachtungen Tuber zweidimensionale regulare Variationsprobleme, Acta Literarum ac Scientiarum Regiae Universitatis Hungaricae FrancisoJosephine, Sectio Scientiarum Mathematicarum, Szeged, 228253', 19241926. Riesz, Marcel
1. L'integrale de RiemannLiouville et le probleme de Cauchy, Acta Math., 81, 1222 (1950). 2. Sur les fonctions conjugees, Math. Z., 27, 218244 (1927). 3. Sur les maxima des formes bilineares et sur les fonctionelles line aires, Acta Math., 29, 465497 (1926). Schauder, Julius 1. t ber lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 38, 257282 (1934). 2. Numerische Abschatzung in elliptischen linearen Differentialgleichungen, Studia Math., 5, 3442 (1934). 3. Der Fixpunktsatz in Funktionalraumen, Studia Math., 2, 171180 (1930). Schechter, Martin 1. Solution of the Dirichlet problem for equations not necessarily strongly elliptic, Bull. Amer. Math. Soc., 64, 371372 (1958). 2. Remarks on elliptic boundary value problems, Comm. Pure Appl. Math., 12, 561578 (1959).
3. Various types of boundary conditions for elliptic equations, ibid., 13, 407425 (1960).
4. Mixed boundary problems for general elliptic equations, ibid., 13, 183201 (1960). 5. General boundary value problems for elliptic partial differential equations, ibid., 12, 457486 (1959).
6. On estimating elliptic partial differential operators in the L2 norm, Amer. J. Math., 79, 431443 (1957). Schwartz, Laurent 1.
The'orie des Distributions, I, Hermann, Paris, 1950.
Shamir, E. 1. Mixed boundary value problems for elliptic equations in the plane. The LP theory. Ann. Scuola Norm. Sup. Pisa, (3) 17, 117139 (1963). Slobodeckii, L. N. 1. Estimates in L,, of solutions of elliptic systems, Dokl. Akad. Nauk SSSR, 123, 616619 (1958) (Russian). Sobolev, S. L. 1. On a theorem of functional analysis, Mat. Sb. N.S., 4, 471497 (1938) (Russian).
BIBLIOGRAPHY
299
2. On a boundary value problem for polyharmonic equations, Mat. Sb. (44) 2, 465499 (1937) (Russian). 3. Some Applications of Functional Analysis to Mathematical Physics, Leningrad, 1950 (Russian).
Stampacchia, Guido 1. Contributi alla regolarizzatione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellittiche, Ann. Scuola Norm. Sup. Pisa, (3) 12, 223245 (1958).
2. Problemi al Contorno ellittici con dati discontinui dotati di soluzioni holderiane, Edizioni Universitarie `f," Genova, 1960.
3. Problemi al contorno misti per equazioni del Calcolo della variazioni, Ann. Mat. Pura Appl., (4) 40, 177193 (1955).
Vekua, I. N. 1. New Methods for Solving Elliptic Equations, MoskowLeningrad, 1948 (Russian). 2. Systems of partial differential equations of first order of elliptic type and
boundary value problems with applications to the theory of shells, Mat. Sb., 31, 217314 (1952) (Russian). 3. The problem of reducing differential forms of elliptic type to canonical form and the generalized CauchyRiemann system, Dokl. Akad. Nauk SSSR, 100, 197200 (1955) (Russian). 4. Generalized analytic functions, Moscow, 1959 (Russian). English translation published by Pergamon Press, London, 1962. Visik, M. I. 1. On strongly elliptic systems of differential equations, Mat. Sb. (71) 29, 615676 (1951) (Russian). Weyl, Hermann 1. Die Idee der Riemannschen Flache, Teubner, LeipzigBerlin, 1923; Chelsea, New York, 1947.
Zaremba, Stanislas 1. Sur le principe du minimum, Bull. Intern. Acad. Sci. Cracow., 197264 (1909).
2. Sur un probleme toujours possible comprenant, a titre de cas particuliers, le probleme de Dirichlet et celui de Neumann, J. Math. Pures Appl., 6, 127163 (1927). Zygmund, Antoni 1. Trigonometrical Series, Monografje Mtematyczne, Warsaw, 1935; Chelsea, New York, 1952; Cambridge, 1959.
SUPPLEMENT I
Eigenfunction Expansions By Lars Garding
Eigenfunction Expansions I. Introduction
The spectral theorem for selfadjoint linear operators is one of the most successful abstractions of mathematics. It has scores of applications to analysis and to classical physics, and it is one of the mainstays of quantum mechanics. Since the theorem was first
proved by Hilbert [10] fifty years ago, it has passed through various stages. The fulldressed version for unbounded operators is due to von Neumann [ 18.1 ], the theory of normed rings has
contributed a very elegant proof [14] and the theorem itself has branched out into a general spectral theory (see, e.g., [6, 9, 12, 20, and 22.2]) and into the theory of noncommutative rings of operators [ 16, 18.2, and 18.3]. We shall be concerned with one simple aspect of the theorem which has been somewhat neglected until a few years ago, namely, its application to singular eigenfunction expansions. There is a historical reason for this neglect. When Hilbert had proved the spectral theorem for completely continuous operators, which have a complete set of strict eigenfunctions, and wanted to
generalize it to continuous, i.e., bounded, operators, he had to find a substitute for the strict eigenfunctions since they no longer
formed a complete set and even could be entirely missing. In modern notation, this substitute was a set {Ex} of commuting projections labeled by a real parameter a and nondecreasing with I. In terms of these projections, the spectral theorem is symbolized by the celebrated formula (1)
A=fAdEA
where A is the selfadjoint operator. It has a sense also for nonbounded operators. Although fairly satisfactory from a theoretical point of view, the spectral theorem in this form is not easy to
apply to concrete cases and provides no immediate way of dealing with the generalized eigenfunctions which appear, e.g., in the theory of the Fourier integral and in quantum mechanics. 303
304
L. GARDING
Since the singular eigenfunctions do not belong to the Hilbert space and the spectral theorem is concerned exclusively with that space, it is clear that some work is required in such an application,
but it can be reduced considerably if we replace (1) by an alternative statement which is implicit in the work of Hahn and Hellinger (sec Stone [22.1]). It is formulated explicitly by von Neumann [18.3] and says that any selfadjoint operator can be diagonalized by a unitary transformation. This is also the usual formulation in modern spectral theory. We shall illustrate it in the finitedimensional case.
The simplest way of stating the spectral theorem for a selfadjoint operator A on a finitedimensional Hilbert space H is to say that A has a complete set of orthonormal eigenvectors. We shall restate it in another form. First let m be the dimension of H and let run through a set Q of m elements, e.g., real numbers and eigenvalues and label the eigenvectors 0 =
2_
A with these numbers. Secondly, let L be the Hilbert space of functions from the set Q to the complex numbers with the scalar product
Then 5EQ
(2)
defines a unitary mapping from H to L such that the operator UAU*
(3)
which transforms L into L, is diagonal in the sense that it is multiplication by a function, in this case A
This is nothing
but a restatement of the fact that
If12=I If,0(s)I2 and
(U.4f)() _ (4f(5)) _ (f4()) _ %()(f,()) _ %()(Uf) so that
(U.4U*F)(E) _ %($)F() if we put F = Uf. The explicit form of U* = U1 is (4) U*F = F()O(E) Hence, via the formula
f = U*(Lrf)
EIGENFUNCTION EXPANSIONS
305
U gives the eigenfunction expansion of any element f of H. We shall call Uf a Fourier transform off. It is important to note that the diagonalization (3) of A is not unique. In fact, a relabeling of the eigenfunctions will lead to a different function A. Additional requirements may be imposed to give unique canonical diagonalizations. They will be discussed below. In the general case of the selfadjoint operator A on a separable Hilbert space H, the spectral theorem is true in the form we have
just stated it, provided we take Q to be all real numbers and let L
be all functions which are square integrable with respect to a suitable measure. The orthonormal set of eigenfunctions of A disappears from the scene, but the unitary mapping U does not, and A is still diagonalized in the sense that UAU* is multiplication by a function. We may say that Uf f gives the Fourier coefficients of F without using a basis, and we shall refer to Uf as a Fourier transform off. The main object of this lecture is to restore the formula (2) by defining suitable generalized eigenfunctions outside the Hilbert space. How this is done will depend on how A and H are defined. We shall see that if H is, e.g., all square integrable functions on
an open subset of real nspace, then (2) and (4) still hold independently of how A is defined, with the eigenfunctions replaced by distributions. When A is an elliptic differential operator, these eigendistributions are functions which are not square integrable in general. Also in the abstract case it is possible to get a complete set of eigenfunctions by making them linear functionals on some suitably topologized linear dense subspace of the Hilbert space H. The choice of this subspace is, however, arbitrary, and the present state of the theory is fragmentary and unsettled.'
Apart from a few remarks, the contents of the lecture are not new; references to the literature will be given as we go along. Before continuing to our main subject, we shall state the spectral theorem in a more precise form without giving complete measure
theoretical details. For simplicity, we limit ourselves to eigenfunction expansions connected with selfadjoint operators. Most ' Added in proof. Since this was written, the introduction of rigged Hilbert spaces has clarified the theory. See Gelbaud and Vileukin, Generalised Functions, Vol. 4, Chapter 1, Moscow, 1961.
L. GARDING
306
of the results carry over to symmetric operators (see [1] for the abstract theory and [5.3 and 17] for the applications). U. Direct Integrals. The Spectral Theorem
Let R" be real nspace and C the set of complex continuous
functions with compact supports in R". We shall deal with Stieltjes integrals
t = (ti,
51(1) dy(t),
. ,
t")
where f is a function in C, or, more generally a Borel function, and u is a measure. We assume that it is a Radon measure, which means that the integral is nonnegative when f is nonnegative and that it is bounded by the maximum of If (t) I times a constant which depends only on the support off. Conversely, by a classical
theorem by F. Riesz, any linear functional on C with these properties is the integral with respect to some Radon measure u. We get a measure ,u of this sort by putting, e.g., du (t)  h (t) dt,
dt = dill
... ,
dt,d
where h > 0 is locally integrable, but the measure can also be concentrated to a countable subset Q of R", in which case the integral is a sum
if (t) P (t) I
teQ
where u(t) > 0 is the measure of the point t. In the latter case we say that It is discrete. We shall denote by LZ(R",y)
the Hilbert space of all Borel functions f from R" to the complex numbers with finite square integral (1)
1112
=f If (1)21 du(t)
and the scalar product (2)
(f,g) = ff(t) g(t) du (t)
EIGENFUNCTION EXPANSIONS
307
Strictly speaking, we get a Hilbert space only if we identify two functions f and g when
If gI =0 or, which is the same thing, when f (t) = g(t) except in a set of ,umeasure zero. More generally, let v be a dimension, i.e., a Borel function from
R" to the integers 0, 1, 2,
... and ac and consider vectorvalued
Borel functions
k = 1, ... ) v(t) where the number of components is v(t) and varies from point to point. If we put
f (t) = (fx(t)),
f(t)
g(t) _ jJJ(t)gx(t) If(t)12
(3)
and
= f(t) '.f(t)
the integrals (1) and (2) have a sense for our new functions and we denote the corresponding Hilbert space by (4)
L2(R",#,v)
In the sense of von Neumann [18.3], it is a direct integral with
respect to y of a collection of Hilbert sequence spaces H, of dimensions v(t). The formula (3) means that we have referred each Ht to an orthonormal basis. This is done only for simplicity. In the last section we shall make use of other bases. We can now state the spectral theorem for a finite number of selfadjoint, bounded, and commuting operators
A,,...,Am on a separable Hilbert space H. It says that they can be simultaneously diagonalized by a suitable unitary' mapping U from H to a suitable
H, = L2(R",,,v) There are many such diagonalizing transformations U and we
can, if we want to, even prescribe that n = 1 and v = 1 (see [18.4]). It is, however, customary to choose a canonical diagonalization by requiring that m = n and that UAkU* is multiplication by the kth coordinate
L. GARDING
308
This means, in particular, that
now gives us the multiplicity
of the composite eigenvalue rr
rr
of the operators A. For canonical diagonalizations there is a uniqueness theorem (see [18.3]) : the measure y and the dimension function v are determined up to equivalence in the sense that this term is used in measure theory.
The spectral theorem extends to nonbounded selfadjoint operators A if we take commutativity in the sense that the bounded operators (A1 ± 1)r, ... , (A. ± i)1
commute with each other. Let us now consider some examples. First of all, we assume that the measure ,u is discrete. Then we have complete analogy with the finitedimensional case: the diagonalizing transformation U can be expressed in terms of a complete set of mutually orthogonal eigenfunctions of the operators A. In fact, let y be concentrated to a countable set Q and construct for every in Q a set of functions *)5
k = 1, .
. .
, v( )
which vanish except at the point E where
Then Fk(E) = (F,G($,k))
for any F in H1 so that (f,g($,k))
for any f in H, if g($,k) = U*G(E,k) is the image of G(e,k) in H. It is clear that the functions G and g form complete orthogonal sets in H, and H, respectively, and are eigenfunctions of the operators UAU* and A, respectively. Conversely, if the operators A have a complete orthogonal set of eigenfunctions, we can, as we did in the introduction, construct a diagonalizing transformation U from H to a suitable space of the form (4) with a discrete measure
ju; if we want to, we can take n = 1 and v = 1. Wellknown examples are the completely continuous operators, e.g., the various classical inverses of Laplace's operator in a bounded region.
EIGENFUNCTION EXPANSIONS
309
We shall now give an example of a diagonalization with a nondiscrete measure. We choose the`Fourier integral
i:rf (x) da
(W) ( ) =J e
(5)
where
da = dxl ... dx It is well known that U is a unitary mapping from H = L2(R",a)
x = 1x1 +
. . .
,
to
H1 =
du = (27T)n dal
... dEn
and provides us with a canonical diagonalization of the differential operators (6)
k = 1, ... , n
Dk = a/iaxk,
They are selfadjoint if we take the domain of definition of D. to be all f such that Dj is square integrable, or, equivalently
JI(ii2d,U < ac The transformation U1 = U* is given by (U*F) (x)
=f
dy
Observe that we can write the right side of (5) as (f,o) where 0 = eiz6 is an eigenfunction of the operators (6) which is not square integrable. We shall meet more examples of this kind of degeneracy of (0.2) in this and the next section. By way of the formula
f (x) = LT*U.f(x) =
f
dy
we get an expansion of an arbitrary element f of H in terms of the eigenfunctions 0. If p(s) = p(EU .
. .
, Sn)
is a polynomial, U diagonalizes the differential operator (7)
B = p(D,i
. . .
, D,a)
L. GARDING
310
If p is real, B is selfadjoint if restricted to functions such that
dµ < 00
We have in this case a noncanonical diagonalization. It is instructive to make it canonical. This can be done by a transformation of variables as follows. Setting aside the trivial case when p is a constant, let t be a real parameter and let St be the

nonsingular (n 1)dimensional part of the level surface p($) = t. Let N be the set of numbers t, for which St is not empty and define for t c N a measure cut in St by putting dtu = dt
dmot
Consider the Hilbert space Ht = L2(St,wt) of functions on St, square integrable with respect to wt, and let k = 1, ... , v(s) be a complete orthonormal set in Ht. If n = 1, is the number of zeros of t and if n > 1, we have to put x. Put H2 = L2(R,a,v) where da = dt on N and 0 otherwise. Finally, define a mapping V from Hl to H2 by putting (8)
(VF)k(t) =f
;F(E)hk(t,E)
when t e N and VF(t) = 0 otherwise. The reader will have no trouble verifying that V is unitary and that W = VU, which is a unitary transformation from H to H2i diagonalizes B canonically. An easy example is furnished by or, in quantum mechanical terms, the nonrelativistic Hamiltonian of n free particles. Observe that the spectrum of any B of the form (7) has uniform infinite multiplicity when n > 1. From (8) we obtain an analogue of (0.2) by expressing F = Uf in terms off. Formally the result is where (9)
( 4Tf)k(t) = (f,4k(t))
Ok(t,x) =Jet
`hk(t,)
 dwt()
EIGENFUNCTION EXPANSIONS
311
If Se is compact, which it is if B is elliptic in the sense that the polynomial p has definite principal part, then S6k is an eigenfunction
of B, but it is not square integrable. When St is not compact, but the functions h have compact supports, we still get functions
0, but for arbitrary orthonormal sets this is no longer true. To get an example of this, we can, e.g., put n = 2 and p = i8E2 with a larger integer s. Parameterizing St with dcul =
we get
28 d$j
so that any function vanishing at infinity and behaving like for small values of $, is square integrable on St. For such a function, however, the right side of (9) is not defined, but it is easy to see that it makes sense as a distribution. In fact, we shall see later that the eigenfunctions of differential operators always exist as distributions. III. Generalized Eggenfunctions
We are going to study situations when the formula (0.2) degenerates without losing its sense altogether. We begin with an interesting halfway abstract case treated by Gelfand and Kostyucenko [8]. First, a few notations. We let S = S(V) be the set of all complex infinitely differentiable functions vanishing outside compact subsets of an open subset V of real nspace. The results are true
also for functions with values in a finitedimensional complex space, but we stick to complex functions for simplicity. Derivatives
will be denoted by Dk = a/iaxk and we put
D. = Dal, ... , D«,n and denote its order by Ial = m. To these symbols we add Da = 1, of order 101 = 0 and the norms max IDJ(x)I,
1«I < m
and
IDmfI = sup IDmf(x)I,
xeV
A subset W of V is called precompact if its closure W is compact and contained in V. We let S' be the space of distributions in V
(see [21]). It consists of all linear functionals L on S with the
L. GARDING
312
following continuity property. To every precompact open subset V' of V there exists an integer m and a constant c such that
IL(f)I
f
(f19) = fg dx,
f,g e s
any locally integrable function g gives rise to a distribution LD (f)
= (f g)
and any distribution is in fact a weak limit of such distributions with g infinitely differentiable. Following L. Schwartz, we identify
L. and g in this case and write
L(f) = (f,L) also in the general case. Now let (f,g) = P (.f g)
be a scalar product in S which is a distribution in V x V. This means that to any given precompact subset V' of V there exists an integer m and a constant c such that I (fg) I< c I D'f I IDmgI
when f and g belong to S(V'). The least integer m for which the inequality holds with some constant c is called the order of the scalar product P. We can, e.g., put
(f,g) = fj b«#DJDflg dt where the coefficients b are locally integrable and all except a finite number vanish outside any given compact subset of V. The simplest choice of P is (.f g) = (f, g) Let H be the Hilbert space obtained by completing S in the norm
(2)
IfI =N/(ff)
EIGENFUNCTION EXPANSIONS
313
let A be a selfadjoint operator on H whose domain of definition includes S and let
Hqf
Uf E H1 =
be a diagonalization of A. Then we have ([8]). For almost all the components of the Fourier transform of a function f in S are linearly independent distributions. Their
THEOREM 1 :
orders have bounds which depend only on the order of P and the dimension n.
We shall sketch a simple proof which uses an idea of Berizanskii
[3]. The measuretheoretical details we leave to the reader. Consider the function
C(x) = (27r)" f e_iz;(l Ex)1 dE,
Ixl
p>n
It is p  n  1 times es continuously differentiable and it is a fundamental solution of the differentiable operator
L = L (D) = Y
DX,
17.1
p
in the sense that
fC(x y)Lf(y) dy =f(x) for any f in S(R). Let V' and V" be open precompact subsets of V which telescope so that V' c V" c: 17" c V, let h c S be 1 on V' and put
b(x,y) = h(x)C(x y)h(y) Then, if f E S(V') we have (3)
f
f (x) = b(x,y)Lf(y) dy
It is clear that if we choose p large enough, then b(,_y) E H
for every}', and an easy approximation argument will show that operator L' commutes with the integration in (3) so that (4)
fB(E,y)Lf(y)
dy
L. GARDING
314
for almost all , where (Ub(,y))($)
This is true in fact for any bounded operator from H to Hl. It is not difficult to prove that the function B can be chosen to be a Borel function in both arguments. Since U is unitary, Ib(,y)12 =
f
da(d)
so that
j'Ib(.,.y)I2d y=f
da(d) dy <
00
This shows that
5IB(,y)I2dY < 00
for almost all $ and hence the components of the right side of (4) are indeed distributions. It is plain that the orders of these distributions in V' do not exceed n + I plus the order of the scalar product P in V". Varying the set V', we find that for almost such that all there exist distributions (5)
(Uf)k() _ (f, ()),
k = 1, 2,
... ,
for almost all E and all f in S. It is not difficult to prove that they
are linearly independent (see [2]). The spectral synthesis of a function f from its Fourier transform F = Uf is given by the formula
Mf =J
da(d)
where M is a mapping of H to S' defined by
(Mfg) = (f g) It is clear from (5) that the distributions 0x play the role of
(6)
eigcnfunctions of A although they do not belong to the Hilbert space H. There are special situations where this statement can be made more precise, e.g., when A maps S into a set of functions
EIGENFUNCTION EXPANSIONS
315
with compact supports which are so smooth that almost all eigendistributions are linear functionals in the set. Then (7)
(Af ,o(ff)) _
feS
which may also be written as
where A' is the adjoint of A with respect to the scalar product (1). An interesting example of this situation occurs when (2) holds
and A is a differential operator with sufficiently differentiable coefficients. If A happens to be elliptic ([4.1, 7.1, and 19]) or hypoelliptic ([11, 4.2, and 4.3]), then (7) implies that 0($) is a smooth function and we get an eigenfunction expansion similar to the classical Fourier transform. It turns out, (sec [8]), that the growth of the eigenfunctions at the boundary of S is limited by the growth of the coefficients of A. We obtain another interesting
application of theorem 1 if we let the operator M in (6) be an elliptic differential operator and let A be given (on S) by another differential operator L so that (Af,g) = (Lff g),
fgeS
(see [8 and 4.3]). Both M and L are supposed to have sufficiently smooth coefficients and to be symmetric in the sense that
( f,Mg) = (Mfg),
(Lf g) _ (f,Lg) In that case there exist vectors V(E) whose components are distributions S such that
(Uf)(E) _
If M' and L' are the adjoints of M and L with respect to the scalar product (1), this means that
L is elliptic or hypoelliptic, tp(E) is a function.
Theorem I was devised to apply mainly to differential operators.
There is another simple result (see [7.2]), due essentially to
L. GARDING
316
Mautner [15] and generalized in [2], which applies to integral operators. Let H = L2(W,a) be the space of square integrable functions on a measure space W with the measure a. We assume that W is a countable union of sets of finite measure and that His separable. We say that an operator B on H has the Carleman property if its domain of definition D(B) is dense and if (Bf) (x)
is a linear functional off e D(B) for almost all x in W. This means that
(Bf)(x) = Ib(y,x)f(,) dot(y)
(8)
where
r
c2 (x) = I I b (y,x) I2 da (y) < oo
so that B has a Carleman kernel b. It is possible to choose b to be measurable on W x W. We let C(B) be the set of functions f in H for which r (9)
I
I f(x) I c(x) da(x) < oo
It is dense in H. .1 Now let A be a selfadjoint operator on H and let
H3f* LfeHI =L2(R,a,v) be a canonical diagonalization of A. We say that a complex measurable function h defined on R is essentially different from zero if the set where it vanishes is the sum of a anullset and a (necessarily countable) set contained in the point spectrum of A. THEOREM 2:
If B = h(A) has the Carleman property for some
function h which is essentially d ferent from zero, then for almost all A,
.(10)
((f)(A) = (f,o(A)),
f c C(B)
where the components of O(A) are linearly independent functions on W. When A is in the point spectrum of A, then
5I(A,x)I2d(x) = fI9k()I2 da(x) < 00
EIGENFUNCTION EXPANSIONS
317
Otherwise
ft(x) I0(A,x)12d% (x) < oo
(11)
for almost all
,
and every function t > 0 such that ft(x)c2(x) dot (x) < oo
Stripped of measuretheoretical detail, the proof of this theorem is very simple. From (8) we conclude that
f
(B*f) (y) = b(y,x)f(x) do, (x)
where f c C(B). It is not difficult to see that the operator U commutes with the integration and this gives (UB*f)(A)
=f B(1.,x)f(x) d«(x)
where B(A,x) = (Ub(x, ))(A)
(12)
Dividing by h we get (10) with
4(A,x) = h'().)B(2,x)
(13)
It is easy to prove that the components of O(2) are linearly independent (see [2]). To get (11), we combine (12) and Parseval's formula, getting
f
c2(x) = I b(x,y) I2 dx(y) = f IB(2,x) I2 dd(.)
so that multiplying by t(x) and integrating
f
ff IB(2,x)12 t(x) dx(x) da(A) = t(x)c2(x)  dx(x) < oo
By virtue of (13), this implies (11) for almost all A.
Clearly, Theorem 2 applies where A itself has the Carleman property and this makes it possible to apply it to any selfadjoint operator A on a separable Hilbert space H. In fact, if we refer
L. GARDING
318
H to a complete orthonormal set we can think of it as a space of functions L2(W,x) where W consists of all integers of arbitrary sign and y assigns the measure one to every integer. If we choose all elements g1, x e W, of the orthonormal set in D(A), then where
f c D(A)
(Af) (x) = I a(x,y)f (y), a(x,y) =
is the matrix of A. Hence, A has the Carleman property, (9) reduces to
I If(x) I IAg.I < oo we have
and
A (A,x) = (Ua(x, ))(A)
(14)
I0().,x)I2
t(x)
< Co
where now t(x) IAgsI2 <
°C'
It is clear that difference operators furnish interesting examples of this situation. The estimate (11) shows that almost all eigenfunctions ck give linear functionals (15)
(ff ck(A))
on suitable subspaces of C = C(B) which are still dense in H. One small such subspace can be obtained as follows: we choose the elements gZ of the orthonormal set in the intersection I of all D(Ap), p = 1, 2, ... , and let Co be the subspace of H characterized by
I If (x) I
I Ag=I °
I APgxl 1(x) < x
for all integers p and q > 0. It is left to the reader to verify that Co is dense in H and contained in 1 and that A maps Co continuously
into CO. The linear forms (15) are clearly continuous on Co and we get (P(A)f ,0k(A)) = p(1) (f k(A)) for every polynomial P, everyf in Co, and almost all A.
EIGENFUNCTION EXPANSIONS
IV.
319
Ordinary Differential Operators
The best understood examples of singular eigenfunction expansions are those connected with ordinary differential operators of the second order (SturmLiouville operators) or, more generally,
ordinary differential operators of any order, and in fact, the first
general theory (Weyl [24]) preceded the abstract spectral theorem by twenty years. The literature on the subject is very large and there are expositions in, e.g., [5.1 and 17]. The fact that we now have a basis for the eigenfunctions makes it possible to obtain very explict results. In particular, if the basis is analytic in the eigenvalue parameter, the diagonalization can be obtained from Green's function via a formula due to Titchmarsh [23] and Kodaira [13]. The majority of authors in this field including, of course, Weyl, rely on complex integration or limiting processes as substitutes for the spectral theorem. For this reason, an abstract approach to the subject still has some freshness. Following [7.3],
we shall give a brief deduction of the main results from the general theory of the preceding section.
Let I be an open interval of the real line, bounded or not, let ..22 be the space of locally square integrable functions on I and Y02 the space of the functions in .72 which vanish outside compact subsets of I. Let
L = amDm + ... + a0i
D = d/idx
be a linear differential operator in I of order m > 0 and
+. + a L* = Dma. m
0
its adjoint with respect to the scalar product (1)
(f,g) = f fg dx
We require that L be formally selfadjoint, i.e., that L* = L. The coefficients of L cannot be arbitrary. We require that at has locally square integrable derivatives of order
320
L. GARDING
We make L into a closed operator from Y2 to 22 by letting
its domain of definition D(L) be all functions in 22 whose derivatives of orders m  1 and m are absolutely continuous and locally square integrable, respectively.
Now let H = L2(I) be the space of all square integrable functions in I with the scalar product (1). If both its domain and its range are restricted to H, L becomes a closed densely defined operator B with the property that
B*cB i.e., B* is symmetric. If the deficiency indices of B* are equal, which we assume, then B has selfadjoint restrictions obtained by imposing linear boundary conditions on the functions of D(B). They have the form
W(.fg) = (L.fg)  (f ,,g) = 0 where g runs through a maximal set M of solutions of B, = ig with the property that W(M,M) = 0. Let A be any such selfadjoint restriction of B and
H3f, Eye H, = L2(R,or,v) a canonical diagonalization of A. By the general theory, there exist for almost all A linearly independent eigenfunctions 0k(A,x), (k = 1, ... , v(A)), solutions of (2)
14k = AOk
such that
(U )k(A) = (f,c k())
when f E 2. In particular, the dimension function cannot exceed the order m of the differential operator L. To prepare the ground for the TitchmarshKodaira formula, we are going to put this result into another form by choosing new nonorthogonal bases for the eigenfunctions. Let (3)
Si(A,X),
. . .
, S.(A'X)
be a basis for the solutions of Ls' = As' whose elements are Borel functions of the pair A, x. If we express the functions k(A,x)
EIGENFUNCTION EXPANSIONS
321
in terms of this basis, Parseval's formula takes the form
(J,f)
J
(fsj(A))(JiSk(A))C,k() da(A)
where (c,k(A)) is a positive semidefinite Hermitian matrix of
rank r(A). More precisely, let H2 be the Hilbert space of vectorvalued Borel functions F(a) = [F1(A), . . . , F,.(1)] with the scalar product (F,, G)
= f .1 F,Gk dP,k(2)
where dp,k(A) = c,k(A) da(%) is the differential of nondecreasing positive semidefinite Hermitian matrix p, called a spectral matrix. Then (4)
H3 f  Vf = ((fs,(1)), ...
)
is a unitary mapping from H to H., such that V.4 V*
is multiplication by a. Clearly, the spectral matrix depends on A and the choice of the basis (3). As we shall see, it can be expressed explicitly in terms of Green's function. In this connection, we have to define Green's function as the kernel of the resolvent
R=(ACE)1,
Iml00
where E is the identity operator and t a complex number. Quite independently of the spectral theorem it is easy to show that
f
(R1f) (x) = g(e x,y)f(y) dy
where the kernel g is a continuous function of the triple f., x, y
except for a jump at x =y when m = 1. It is analytic in / and square integrable in one of the variables x and y when the other is fixed. Further g(I', y,x) = g(t,x, )')
(5)
and (6)
(Ly
 C)g
= (L.  ()g = 0
L. GARDING
322
when x y. For x =y, Green's function has the classical singularity: for fixed x and (, the function Dm'g is continuous except
at the point y = x where it has an appropriate jump. It also satisfies the boundary conditions that determine A in the sense that hg e D(A),
(7)
(/and y fixed)
if the function h e D(L) vanishes in a neighborhood ofy and equals
1 in a neighborhood of the endpoints of the interval I. Most of these properties hold also when A is a selfadjoint restriction of an elliptic partial differential operator L with suitably differentiable coefficients defined in an open subset of the real ndimensional space. The resolvent is still given by a kernel g(e,x,y), continuous
in the triple C, x, y when x y and Im t 0 and analytic in The formulas (5), (6), and (7) are still true, but the singularity has to be described otherwise. In a neighborhood of the diagonal x =y, Green's function behaves like rm", where r is the distance between x and y, or when m  n >_ 0 and n is an even integer like r"" log r1. It is a Carleman kernel if, and only if, 2m  n > 0.
Returning to ordinary differential operators, we can now connect the spectral matrix with Green's function. First of all we choose the basis (3) in such a way that all its elements are entire analytic functions of A. This means that they are defined when A = /,is complex and
Ls =fs for every I. We get such a basis, e.g., by solving an appropriate
Cauchy problem. Since the functions s are now analytic, it follows easily from (5) and (6) that Green's function has the form
g((,x,y) _ I Mk(f)S!(l,x)sk(fy),
yzx
g(fsx,y) = I Mi
vSx
(f)si(f,x)sk(/ y),
where sk(C,y) = sk (l,y) and the functions A11(1)
are analytic off the real axis. Let M(f) be the matrix whose elements are
M,k(P) = Al (f) I Mix (10)
EIGENFUNCTION EXPANSIONS
323
Then we have the TitchmarshKodaira formula M(1)
(8)
_
dP(A)
J
Af
which should be taken in the sense that rN
M(f) 
dp(A) ,
N>0
is analytic across the real axis in the interval N, N. More explicitly, this means that p(A) = 7r1 lira lim
ra+a I
e. +0 d
(M(,u + ie)  M(,u  ie)) dµ/2i
if we normalize p so that it is continuous from the right and p(O) = 0. That (8) holds, is a fairly easy consequence of the formula
ffg(t,x,y)f(x)f(y) dx dy = (R efj )
= f (A 
8)i Y (fss(A))(fsrk(;)) dAJk(A)
which in turn follows from the spectral theorem.
Bibliography 1.
N. I. Aehieser and I. M. Glazman, Theorie der linearen Operatoren im Hilbertraum, Berlin, 1954.
2. 3.
W. G. Bade and J. T. Schwartz, On Mautner's eigenfunction expansion, Proc. Nat. Acad. Sci. 42, 519525 (1956). Yu. M. Berezanskii, Eigenfunction expansions of partial differential and difference equations, Proc. Third Congress of Soviet Math., 3637 (1956).
4.1. F. E. Browder, The eigenfunction expansion theorem for the general selfadjoint singular elliptic partial differential operator I, II, Proc. Nat. Acad. Sci., 40, 454463 (1954).
4.2. F. E. Browder, Regularity theorems for solutions of partial differential equations with variable coefficients, Proc. Nat. Acad. Sci., 43, 234236 (1957).
4.3. F. E. Browder, Eigenfunction expansions for formally selfadjoint partial differential operators I. II, Proc. Nat. Acad. Sci., 42, 769771, 870872 (1956).
L. GARDING
324
5.1. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrawHill, New York, 1955. 5.2. E. A. Coddington, On selfadjoint ordinary differential operators, Math. Scand., 4, 921 (1956). 5.3. E. A. Coddington, On maximal symmetric ordinary differential operators, loc. cit., 2228. 6.
J. M. G. Fell and J. L. Kelley, An algebra of unbounded operators,
Proc. Nat. Acad. Sci., U.S. 38, 592598 (1952). 7.1. L. Girding, Eigenfunction expansions connected with elliptic differential operators, The Twelfth Congress of Scand. Math., Lund, 1953, pp. 4455.
7.2. L. Girding, Applications of the theory of direct integrals of Hilbert spaces to some integral and differential operators, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 1954.
7.3. L. Girding, Kvantmekanikens matematiska bakgrund, mimeographed lectures (in Swedish), Lund, 1956. 8.
9. 10. 11.
I. M. Gel'fand and A. G. Kostyucenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.), 103, 349352 (1955). P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1951. D. Hilbert, Grundzuge einer Allgemeiner Theorie der linearen Integralgleichungen, B. G. Teubner, Leipzig, 1912.
L. Hbrmander, On the theory of general partial differential operators, Acta Math., 94, 161178 (1955).
12.
J. L. Kelley, Commutative operator algebras, Proc. Nat. Acad. Sci.,
13.
K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Am. J. Math., 72, 504544
U.S., 38, 598605 k 1952).
14.
(1950). L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York, 1953.
15.
F. Mautner, On eigenfunction expansions, Proc. Nat. Acad. Sci., 39, 4953 (1953).
16.
F. J. Murray and J. von Neumann, On rings of operators I, II, Ann.
17.
Math. (2), 37, 116229 (1936) ; 41, 208248 (1937). M. A. Naimark, Linear Differential Operators, Moscow, 1954.
18.1. John von Neumann, Allgemeine Eigenwerttheorie hermitscher Funktionaloperatoren, Math. Ann., 102, 49131 (1929). 18.2. John von Neumann, On rings of operators, III, Ann. ,Bath. (2), 41, 94173 (1940). 18.3. John von Neumann, On rings of operators. Reduction theory, Ann.Math. (2), 50, 401485 (1949). 18.4. John von Neumann, Ueber Funktionen von Funktionaloperatoren, Ann. Math. (2), 32, 191226 (1931). 19. A. Ya. Povsner, On the expansion of arbitrary functions in characteristic functions of the operator Au + cu, Mat. Sb. N.S., 32, 109156 (1953).
BIBLIOGRAPHY
325
I. E. Segal, Decompositions of operator algebras, Afem. Am. Math. Soc. No. 9, 166 (1951). 21. L. Schwartz, Theorie dec Distributions, I, 11, Paris, 1951. 22.1. M. H. Stone. Linear transformations in Hilbert space, Am. Math. Soc. Coll. Publ. (1932). 22.2. M. H. Stone, Boundedness properties in functionlattices, Can. J. Math., 1, 176186 (1949). 23. F. C. Titchmarsh, Eigenfunction Expansions Associated with Second Order 20.
Differential Equations, Oxford, 1946. 24.
H. Weyl, Ueber gewohnliche Diferentialgleichungen mit Singularitaten and die zugehorige Entwicklungen willkurlicher Funktionen, Math. Ann., 68, 220269 (1909).
SUPPLEMENT II
Parabolic Equations By A. N. Milgram
Parabolic Equations In this lecture we shall discuss in a general way some of the methods leading to proofs of existence and uniqueness of solutions
of boundary value problems for parabolic partial differential equations. At first we shall restrict our attention to equations of the second order. General methods applicable to equations of order higher than the second can be exhibited just as well for second order equations. Moreover certain relations (e.g., the maximum principle) hold for equations of the second order which do not hold for equations of higher order. The first three sections will describe three different approaches to solutions of the heat equation Au  au/at =f. The first will be a direct Hilbert space approach, the second the method of potentials or integral equations, and the third will be Perron's method.
We shall limit our attentiion only to Dirichlet type boundary
problems. The methods can be adapted to equations with coefficients which are functions of space and time. In the fourth section we shall discuss the interrelation between these methods and also some problems whose investigation might be fruitful.
The fifth and concluding section consists of some remarks concerning parabolic equations of order 2m, m > 1. We call attention here to an obvious gap in the existent general theory. Reference in the sequal were chosen more or less at random to illustrate subject, without considerations of priority or even best results. I.
Direct Hilbert Space Approach to Solutions of the Heat Equation
We describe in this section a proof (based on a method of Vishik') of the existence of solutions of (I1)
Au +
at =f(X,t)
1 M. I. Vishik, Mixed boundary problems, Dokl. Akad. A'auk SSSR, 97, 1936 (1954). 329
A. N. MILGRAM
330
where
0=
a2
axe
+
...
+
a2
and
x = (x1, ... , x,,)
a n,
0
The solution u(x,t) is sought in a domain D, where D = 0 x [0 < t < T] and 0 is a domain in Cartesian nspace En. Denoting by a0 the boundary of 0, we require further that u(x,0) = uo(x) and 0 for e aO, where uo(x), the initial value, is a given function. Let C* denote those functions of class C°° in D which are zero in a neighborhood of the upper boundary of D, i.e., in a neighbor
hood of the lateral surface aO x [0 < t < T] and the of "top," 0 x [t  T]. For simplicity all functions are assumed to be real. for each 4 = O(x,t) E C* we denote ¢(X,0) by 0o. Multiplying I1 by 0 and integrating over D, we obtain (12)
LO(U)
=[

aO
u] + [0,u] i = (0o,uo) + [O,,f ]
where the symbols in (12) are defined as follows: (go)uo)
=LOO(x)uo(x) dxi
[/,u] = (I3)
u Y= (0o)o)
... dxii T
D
oO(x,t)u(x,t) dx2 ...dxn dt
 u dx dt = o
auo
n Y1=1
axi axi
fol, (0,u)1 dt
dx 1
...
dxn
foT J(..) dxi ... dx dt i
1100112 We denote the associated norms by 1100112 = = 110112 11011 In addition we define = o = [010], and (0o) 0o) 1, for 0, ,u e C* an inner product
(O,t,)* := (0o' 'o) + [O"v]i
and complete C* to a Hilbert space H* by means of this inner product and norm 110 11 * =
(,)

PARABOLIC EQUATIONS
331
Recalling the wellknown inequalities II0II0 < A 11011 (15)
II0II0 < A II0II1
where A is a constant depending only on 0, we apply Schwartz's inequality to (I2) and obtain ILO(V')I s K(/) (II+'IIo + IIVII1) <_ K(0)[1 + A] IIjpQI*
L,4(V) is considered here to be a linear functional in y, E C* for
fixed, and K(0) is a constant depending on 0. By the FrechetRiesz theorem there exists To E H* such that (I6)
L4(ip) = (V, TO) *
T¢ is clearly a linear operator with domain C*. Taking y, _ (I6), we have
in
(,T)*
= L4(o) = (0o,0o) + [0,0J = 11011 *2 And this combined with Schwartz's inequality (O,TT)* s II0II* II TTII* gives II Toll* >_ II0II*
Thus T is a 11 mapping and has a bounded inverse T1 of norm
<1, which may be regarded as extended to the closure of the range of T. We now consider the right hand side of (I2). Let (I7)
B,,O,f (0) = (c6o)uo) + [0,f J
Evidently,
f
IBuo, (O) l <_ K1 [1100110 + II0 II0]

K2 11011*
where Kl = II UoIIo + llf IIo, and K2 = 2  K1  max (1,A). Not
only is B01() a bounded linear functional for all 0 in H*, but also Buo, f (Tltp) is a bounded linear functional on the closed subspace M of H*, and hence (I8)
BUo.f (T 'V) = (', U) *
for some U E M, and arbitrary V E M.
A. N. MILGRAM
332
Replacing T'y, by 0 in I8, TO, U)
BUa.f
and by (16)
(TT,u) * = L (u) = B,,.f (0) = (00,uo) which establishes t% as a weak solution of (I1). It is worthwhile observing that by replacing y, by u in (18), we obtain 1jul1 *2=B«a.f(T1u)
and hence II UII * c 2 max (1,A)
(19)

(Il U0110 + Iif Ilo)
If in equation (I1) u is replaced by e+"v, the equation assumes the form (I10)
Acv
 iv  at = e'(f (x,t) = g(x,t))
For equations of this form it is evident that II4II* should be and then the preceding discussion will apply without change. If now we consider a function
defined by the relation 11 'I *2 = 110'12
f =f
(txu,
I
u =f (1,
i 110112 + 1!011 ,
au
`
au
ate ,
,
.
,
which satisfies a uniform Lipschitz condition
f
all
t,X,u, ax)
f
(t,x,v,
aL'
ax)
AO I U  vI 
A,
au
av
ax,
ax,
it is readily seen that (I11)
exf (t,x,e I Atu,e,u au
ax)
 e; f
( . [III,
Al
ar

where the constant is independent of A. In the equation (112)
.1u
av
u, =f (t.x,u, au)
V112
2 110
it U 
ti112 2]
PARABOLIC EQUATIONS
333
replace u by e' A  v, obtaining
Av  Av + vit = exf I t,x,zveAt,eA1
(I13)
ax1
It is readily seen that by successive substitutions the functions defined by vo(x,t) = euuo(x)
a,.,
Ov1 } Av1 ' ati = e"f
will converge in *norm (for sufficiently large A) to a solution of I13 and hence after substitution of v = e  Atu, we obtain a solution of 112 with prescribed initial data and 0boundary values. The requirement of a Lipschitz condition in u and au/ax imposed on f (t,x,u, auf ax) can be weakened somewhat. H.
Potential Theory for Parabolic Equations
The method of potentials was applied to parabolic equations many years ago.2 The fundamental solution of the heat equation Z(x,t,y,r) = a(t  T) where
1x12 = xi 
X2
n/2eIrsI2/4(t
 ... + x
7)
and a =
2nir,,12
plays a role in this approach analogous to the role of Ix A "2 in the Laplace equation. For parabolic equations of second order with coefficients depending on space and time the fundamental solution was investigated by F. Dressel (Duke Journal, 1940 and 1946).
Observing the relations
J0
Jf van
y,T) dy dT =
.fo
f
aT
Z(x,t,y,T) dy dr
d6; dT =  f Z(x,t,y, t  e) dy +J Z(x,t,y,0) dy
2 Cf. Gevrey, Journal de Math., 1913.
A. N. MILGRAM
334
and passing to the limit as a k 0, we obtain
where
Z(x,t,,T) dodT = E(x) + f Z(xt0) dy
£JO
o
E(x) = 1 if x c O
IifxE6and
is smoothatx
These are the basic jump relations. Continuing the analogy with classical potential theory, set
U(x,t) = rt
f6
dd, dt
an
U(x,t) is a solution of the heat equation with initial value 0 at be a given function defined on all interior points of 0. Let
the boundary O of 0. The requirement lim U(x,t) = z ' o
where x approaches through interior points, after an application of the jump relations, gives rise to the following integral equations, for (µ00,t) :
A(Eo,t) = [E(go)  1] µ00,t) +f00 J
U(E,T)
!n Z($o,t,$,T) dog; dt
Si nce this integral equation is of Volterra type, a discussion of existence and regularity of its solutions can be achieved by known methods, providing the boundary is sufficiently regular. Naturally
the question of what is meant by sufficiently regular can be an interesting and delicate problem about which we have yet to hear the last word. Also the formulation of minimal assumptions on the coefficients of the second order operator is still a subject of investigation.
111.
The Maximum Principle and Perron's Method
Let E x J be the Cartesian space of n I, 1 dimensions. Denote points of E x J by (x,t) = (x1) ... , xn, t) and let D be a bounded
domain in E. x J. If v(x,t) is a solution of Au = aul at, it is not
PARABOLIC EQUATIONS
335
difficult to establish the relation
(III1)
u(xo,to) S sup u(x,t),
(xo,to) e D
xjeaD
where aD denotes the boundary of D. If D is a cylindrical domain,
D = 0 x [0 < t < T], the assertion (III1) remains valid for Xo e 0, 0 < to < T if (x,t) in sup U(x,t) ranges over the lower boundary of D, i.e., the combined lateral surface aO x [0 < t < T]
and bottom 0 x [t = 0]. In the light of this maximum principle, it is natural to ask whether Perron's method, used first to obtain solutions of the first boundary problem for the Laplace equation, is also applicable to the heat equation.
After defining subparabolic and superparabolic functions, Sternberg,3 Petrowsky,4 and others showed that for each continuous function f defined on aD there exists a solution u of the
heat equation such that U(P) =f (P) at each boundary point P e aD around which there exists a barrier. The problem of determining nontrivial sufficient conditions for the existence of a barrier is fundamental in such investigations.
A function 0 defined in a domain D of E x J may be called superparabolic if (1) 0 is continuous in D except at the points of a
finite number of hyperplanes t = constant, and (2) if C is any cylinder with axis parallel to the t axis and contained in D and if u is any solution of Au  Ut = 0 in C such that U < on the lower boundary of C then u satisfies the same inequality, u interior to C.5 Reversing inequalities yields the definition of subparabolic functions.
A barrier at a point P0 of D is a superparabolic function u,(P) defined for some r > oo in D n S(Po,r), where S(Po,r) is the sphere with center P0 and radius in E,, x J, satisfying the conditions (1)
lim co(P) = 0 and (2) P'Pt
inf
w(P) = CE > 0 for each E > 0.
fP Pol ? e
3 Sternberg, Math. Ann., 101, (1929).
4 I. Petrowsky, Zur ersten randwertaufgabe der warmeleitungsgleichung, Compositio Math., 1 (1935). b This definition is contained in an unpublished paper of Professor W. Fulks, Department of Applied Math., University of Minnesota. See also, J. L. Doob,
A probablity approach to the heat equation, Trans. Amer. Soc., 80 (1955).
A. N. MILGRAM
336
The solution to the boundary value problem is sought as the infimuin of supcrparabolic functions with boundary values > a given function]. In his treatment of the above problem, Petrowsky restricted himself to the case of one space dimension: a2u/ax2 = au/at, and considered a domain bounded by two curves
x=0i(t)
X = 02(t),
0
He proved that there exists a solution with boundary value a preassigned continuous function f given on the base and sides of
D if the boundary curves have the property that for each t, 0 < t < T, there exists a function p(h) (= p(t,h)) such that: (1)
p(h) 10
(2)
01(t { h)
 41(t) >  %Yh log p(n) 02(t  h)  02(t) < Vih log p(h) E
(3)
as hf 0
p(h)'Vi
J
g p(h) I
A   cc
as e ) 0 and C < 0
If (1) and (2) hold but (3) fails, counterexamples can be constructed. These results represent an improvement on Gevrey's condition of Holder continuity on the functions 01, 02 with an
exponent a >  . Such refined conditions have not been obtained for the higher dimensional case. A crude sufficient condition for the existence of a barrier at a point PO E D5 is that PO be the vertex of a cone in the complement of D along the axis of which the t coordinate of a point decreases as the point recedes from PO (see figure above).
The problems discussed in the preceding sections have one feature in common in that all seek solutions of Au = U= which assume prescribed boundary values on D the boundary of a general domain in spacetime. The solution sought cannot in general achieve all prescribed boundary values. For example, if IV.
PARABOLIC EQUATIONS
337
D is a cylinder with axis parallel to the t axis, the boundary values assigned at the top of the cylinder will not influence the solution
if the solution conforms to preassigned values on the lower boundary. Thus the boundary value problems considered must be regarded as a search for solutions of Au = u, which in some sense minimizes the difference between a prescribed function f and the values assumed by u on D. The word "minimize" as used in Section III means equal on a boundary set of "greatest measure." But if minimize were to mean the solution u(p) = u(x,t) is sought
so as to minimize the integral f If(p)  u(p)12 dS taken over the boundary, a new class of problems would arise. In such a problem prescribed boundary values on the top of a cylindrical domain would have material influence on the solution. A question related to the above which might be asked is the
following: let R be a subdomain of D. Let f be a prescribed function on R the boundary of R. What boundary values should be assigned on aD so that the solution u(x,t) best approximates f
on aR ? We leave open the sense in which the words "best approximates" are used. Of the methods used in Sections I and II we can say that the first is modern, the second classical. In the direct Hilbert space approach a solution of the boundary value problem is obtained by first solving Au  au/at = f with 0 boundary values. Then using f =  Af0  afol at, where fo is a function defined and belonging to C2 in D, the function v = u + f0 provides a solution of Av aav/at with prescribed boundary values. The listener will observe at once that this seems to limit the boundary values to those which admit an extension into the interior of D of class C2(D). This would indeed be a very serious defect in the Hilbert space approach. Fortunately, in the case of second order equations the maximum principle comes to the rescue by guaranteeing that a sequence of solutions with uniformly convergent boundary values is uniformly convergent in the interior and the limit is readily shown to be a
solution. Hence the fact that the Hilbert space method is not directly applicable to all preassigned boundary values is not of vital importance. For equations of order higher than 2 the situation is different.
A. N. MILGRAM
338
V.
Equations of Higher Order
Let D be a domain in E" x J, and a = Jti, a2, ... , xK a sequence of integers where 1 < a, < n for each v. Set lal = K $12, ... , 41K. The and D" = aK/(axai 4,2 . . . 4,K), equation
a xt D"u =
V1 jaj:z_ 21n
"
au at
will be called parabolic if Z a,D" is a strongly elliptic operator, i.e., if there exists a constant A > 0 such that
Rt[,ja.(x,t)e] z (1),,1+1A
for2, ... ," real and
I
IEI2"
I2 = ($1)2 + ...
($1)2
0,
and (x,t) E D. Systems of equations can also be treated; definitions will be found in the other lectures. Various methods for treating
(V1) have been given. The most direct and comprehensive is probably the method described in Section I, suitably modified. Of course, the solution obtained is at first only known to be a weak solution, this difficulty is easily overcome. Using the fundamental
solution whose existence and regularity was established (for systems) by Eidelman6 it is easy to verify by known methods that the weak solution is regular. Other methods including Friedrich's mollifiers can be used to verify regularity of the weak solutions in the interior. However the requirement that the boundary values can be extended into the interior, i.e., assumed by a function fo
which has finite mnorm, cannot this time be eliminated by a simple limiting process. For higher order equations no maximum principle is known. Thus even for elliptic equations the general existence theory based on Hilbert space methods is inadequate
to prove, say, the wellknown theorem that the biharmonic equation L2u = 0 admits a solution u in the interior of a sphere so
that u = g, au/ an =f for arbitrary continuous functions f and g on the boundary. It is likely that at least for sufficiently smooth boundaries, the method analogous to that discussed in Section II will enable us to 6 S. D. Eidelman, On the fundamental solution of parabolic systems, Mat. Sb. N.S., 38, 80 (1956).
PARABOLIC EQUATIONS
339
obtain more complete results. Recently R. Juberg (Thesis, University of Minnesota) observing that for the fundamental solution (Eidelman) similar jump relations continue to hold, derived integral equations analogous to those described in Section II. The resulting system of equations, however, are complicated,
and his results are definitive in certain special cases. It is clear that an investigation of the integral equations both for parabolic and elliptic equations in an effort to obtain a Poisson integral type representation of the solution must lead to new and deeper insight into the boundary value problem for parabolic and elliptic equations.
Index
A priori estimates, 231, 263, 286, 289 Adjoint operator, 138 Analyticity theorem, 136, 139, 207
Direct integrals, 306 Dirichiet and Neumann problems, 263, 286
Dirichiet principle, 164 Dirichiet problem. 141, 152, 153, 154, 155, 159, 190, 196, 197, 237, 263, 282, 329 for the difference equation, 157 Discontinuities, propagation of, 53 Distribution, 138 Duhamel's integral, 17, 63
BanachSaks theorem, 181 Beltrami equation, 257, 267 Boundary conditions, 141 Bicharacteristics, 58 Bounded linear functional, 179
CalderonZygmund inequality, 224, 245
Cauchy and Kowalewski theorem, 46 Cauchy data, 44, 62 Cauchy problem, 18, 45 standard, 63 uniqueness of, 47 CauchyRiemann system, 257 Characteristic form, 43, 70, 135 Characteristic lines, 7 Characteristic matrix, 43 Characteristic surface, 21, 43 Classical solution, 137 Closed linear subspace, 178
Elastic waves, equation of, 74 Elliptic, 143 Elliptic equations, 134, 135 Elliptic operators, 143 Elliptic systems, 255 Energy integrals, 24
Coerciveness, 201
Compact support, 7 Complete, 178 Completely continuous, 183,199 Complex notations, 256 Continuity method, 238, 285
Finite differences, solution of parabolic equations, 109 stability of difference schemes, 115 Fourier transformation, 65 Fredholm pair, 142, 175 FredholmRieszSchauder theory, 183
Descent, method of, 16
Free surfaces, 43 Function spaces, 216
Differentiability theorem, 138,
Estimates "up to the boundary," 235 Finite differences, 155 Finite differences, method of, 108
see also Symmetric hyperbolic systems
Fundamental identity, 211 Fundamental solution, 214, 215, 333
139,
174 341
INDEX
342 GArding's inequality, 170, 198
Liebmann's method, 159
General boundary value problems,
Linear equations with wave operator as principal part, 31
142
Generalized derivatives, 138 Generalized eigenfunctions, 305, 311 Gradient vector, 38 Green's function, 215 Green's identity, 48, 104
Heat
conduction,
boundary initial
value problem for rectangle, 104 equation of, 94 initial value problem, 98 maximum principle for, 96, 97 smoothness of solutions, 101 uniqueness of solutions, 96, 97 Heat equation, 329 Hilbert space, 177 Hilbert space approach, 329 Hilbert space methods, 165 Holder condition, 136 Holder continuity of derivatives, 136 Holder continuous, 138 Holder KornL ich tensteinGiraud inequality, 223, 244 Holmgren, uniqueness theorem of, 47 Huygens' principle in strong form, 15 Hyperbolic equations, linear, 69 solution by plane waves, 72 standard Cauchy problem for, 63 wave equation, 1, 2
Hyperbolic systems, Green's identity for, 48 with constant coefficients, 70
see also Symmetric hyperbolic systems
Integral equation approach, 215 Integral equations, 329 Interior estimates, 232 Interior regularity, 190 Interpolation lemmas, 218, 219, 250 Laplace equation, 134, 136 LaxMilgram lemma, 180, 199 LeraySchauder method, 285
Lions, 202
Maximum principle, 150, 160, 262, 334
Maximum principle for heat equation, 96
Maxwell's equations, 87 Minimal surfaces, 282 Mixed boundary problem, 143 Mollifiers, 145
MorreyNirenberg, 210 µconformal mapping, 267 Multiindex, 135 Negative norms, 200 v. Neumann condition, 118 Neumann problem, 154, 283 Norm, 135 Normal boundary conditions, 142 Normal velocity, 23,61
Oblique derivative problem, 154 Ordinary differential operators, 319 Parabolic equations, 2, 94, 329
see also Finite differences and Heat conduction equation Parseval's identity, 65 Periodic distribution, 168, 169, 174 Perron's method, 329, 334 Plane waves, decomposition into, 72 Poisson equation, 134, 137 Potential gas flow, 282 Potentials, 333 Potential theory, 211
Principle of contracting mappings, 283
Privaloff's theorem, 264, 267, 279 Projection theorem, 164, 177, 179 Propagation of discontinuities, 53 Properly elliptic, 144
Quasiconformal, 258
343
INDEX Regularity at the boundary, 200
Representation theorem, 164,
179,
259
RieszSchauder theory, 164 Runge property, 140
Scalar product, 178 Schauder estimates, 231, 238 Schauder's fixed point theorem, 286 Schwarz inequality, 166 Secondorder equations, 150 Selfadjoint operators, 303, 305 Single hyperbolic equations, 62 Singular kernel, 222 Smoothness of strong solutions, 240 Sobolev and Kondrashov, 221 Sobolev's inequality, 220, 242 Sobolev's lemma, 30, 77, 167 Spaces Ht, 165, 167 Spacelike surface, 23
Spectral theorem, 303, 304, 305, 306, 307, 308 Stability of difference schemes, 116 Strictly hyperbolic, 72 Strong derivative, 137 Strong solution, 138 Strongly elliptic, 143, 144 SturmLiouville operators, 319 Subspace, 178
Support of a function, 7 Symmetric hyperbolic systems, 86, 87 difference methods for, 117
Tangential operator, 215 Test functions, 137 Timelike surfaces, 23 TitchmarshKodaira formula, 320
Uniformly elliptic, 150, 255 Unique continuation, 162 Unique continuation property, 140 Unique continuation theorem, 262
Uniqueness, Holmgren's theorem of, 45
Uniqueness for heat equation, 97 Uniqueness theorems, 283
Wave equation, d'Alembert's solution for, 5 characteristics, 21 characteristic cone, 21 characteristic lines, 7 characteristic surface, 21
domain of dependence, 7 Huygens' principle in strong form, 15 inhomogeneous, 17, 18, 19 initial value problem for, 6, 10 mixed problems for, 35
mixed problem for finite x interval, 9
mixed problem for semiinfinite x interval, 7 onedimensional, 4 spacelike surfaces, 21 threedimensional, 10 timelike surfaces, 23 twodimensional, see Descent, method of
Waves, see Plane waves and Elastic waves
Weak derivative, 137 Weak equals strong, 144 Weak solution, 138, 198 Weak vs. strong, 137 Wellposed problems, 2
Zeros of elliptic systems, 261
NBSI E6h009T290
6
Z64008u1Z808L
SyjV uo ayi gaA&
L'£/Wb1
9JOSUQE'AAAMAi