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p. Here, we take the radius p of the sphere greater than / x - I. Tn contrast with (2.10), we now place the center of the sphere at the point x rather thanx. Performing the same transformations a s in the preceding case, we obtain the formula
x
x
Let u s subtract Eq. (2.11) from (2.10) and let us transform the result as follows:
X"[
1
dyi dyi
Ix
-y
In-'
-
I x-
y
128
LINEAR EQUATI ONS
The t e r m s in the braces cancel each other out, as is easily seen.
(2.12)
Here, p is an arbitrary number greater thanlx - 21. With the aid of this representation, we can bound the quantity 1D2wl(m), En in t e r m s
of En just as was done above for vxiXn(x'. xn)-vXlXn(x'.%,). We shall not repeat the details. F r o m Lemmas 2.1 and 2.2, we derive l e m m a 2.3. I f the functions f(x) and Q ( X ) are equal to 0 outside the sphere 1x1c k , then, for the solution U ( X ; of the Dirichlet problem Au =f (XI. u Ixn=0 = 'p (x'). (2.13)
in the half-space x,, >0, which converges to 0 as I x I + 00, the following inequality i s valid:
where the constant c depends only on n. Let u s represent the solution u ( x ) a s a sum of potentials. To do this, we extend f ( x ) onto the half-space x , ,< 0 a s an even function, so that / ( x ' , - x , , ) = f ( x ' . xn). We keep our former notation f o r this extended function. Obviously, Ifl(@), ( X n > o l - If[(=), En. Fromf(x), let u s construct the Newtonian potential
129
SCHAUDER'S A PRIOR1 E S T I M A T E
The function v ( x ) = u ( x ) + w (x) will satisfy Laplace's equation and, for x n = 0, it becomes the function p ( x ' ) w ( x ' , 0). It approaches 0 as 1x1 +me Therefore,
+
21
(4= -
JK
(Jd
-Y) I Y (Y')
Yn= 0
+
'w (Y'l
0)l dY'1
and the solution u (x) is v ( x ) -w (x). On the basis of Lemmas 2.1 and 2.2, inequality (2.14) is valid for u ( x ) . This completes the proof of Lemma 2.3. An arbitrary function u ( x ) of compact support belonging to C2,(En)can be represented in the form of a Newtonian potential (I
with finite density Au. Therefore (cf. Lemma 2.2), for this function, (2.15)
Let u s now turn to the derivation of inequality (1.11). Suppose that u ( x ) E C,,a(a), that S E C,,., that the coefficients L belong to Co, and that 'p (s)= u Is E C,, (S). Let us take a finite number of indefinitely differentiable nonnegative functions (x), ., ,C ( x ) of compact support such that, for every x E
(a),
(L
a,
N
[,
..
xEG.
xcC,(x)=l.
k=l
(2.16)
with the property that the diameters of their supports do not exceed some small number 6. The value of 6 will be made more precise below. It is determined only by quantities that we know. h accordance with (2.16), the function u ( x ) can be represented in the form N
2 U h ( x ) = u (x)s k=1
where
uy ( x ) =sL u ( x ) Ck (x).
Let us denote LU by f(x). Let us multiply the equation Lu = f by C, and then represent it in the form MU
where Fk
=f c k
+
a i j (x) ~
a i j (2uxibX,+
k
I /
=~F , (x). . ~
uC&xiX,)
- aiuxiCk - a d , .
(2.17)
130
LINEAR EQUATIONS
If the support Q, of the function C,(x) is contained in M, and if ' b ( x ) to all En by assigning it the value 0 outside Q k , we can regard u k (x) a s a function of compact support in C2,~ ( E , that ) satisfies Eq. (2.17) with inhomogeneous t e r m F , ? ( x ) . On the other hand, if the carrier 9 , is only partly contained in Q , we set up in it n, in such a way new regular coordinates y, = y, ( x ) , for i = 1 , that the portion S , of the boundary S that belongs to '2, will be represented by the equation y, = 0 and the region Q , n Q will belong to the half-space y, 0. The functions y, ( x ) must be elements of C2,(2,). In the new coordinates, the function (y) = uk ( x ( y ) ) will satisfy an equation of the same form (2.17), namely,
we extend
...,
>
uk
M u , =&/ (Y)
'kYiY,
= F&(Y),
(2.18)
where &,](y) and F , ( y ) are calculated from a l j ( x ) and F , ( x ) by a familiar procedure.Let u s extend u k (y) to the entire half-space y, > 0 by setting ' k (y) = 0 for values of y in (y, 0) that do not belong to the image of the region Q k n 8. We keep the same notation as before for the function i k ( y ) thus extended. In the region y,> 0, it satisfies Eq. (2.18) and, for y, = 0, it satisfies the boundary condition
>
(2.19)
(y) lV,,=O = G k (y')*
where ' 9 k (s) = '9 (s) [ k (s) and .9, (YO = y k (S (Y) 1. Let u s represent Eqs. (2.17) and (2.18) in different forms: Mo', = atj ( x " )U k x i x , = Fk ( X I
[ a i j (x')
-
( X ) ] Uhxix,
(2.17 ')
and (2.18')
where xu and y" are arbitrary points in M k and G k respectively. Instead of x and y, let us introduce new rectangular coordin?tes z with ?rigin at the points x') andybin such a way that Eqs. (2.17 ) and (2.18 ) will be transformed respectively into a;j (o)ukZi~,~4'&=F;(z)+[n~j
(2.17")
(o)-a~j(z)]'Lkz,z 1 /
and
> >
and in such a way that the half-space (y, O } in the second case will be transformed into the half-space ( z, 0).
131
SCHAUDER‘S A PRIOR1 ESTIMATE
Let us apply inequality (2.15) to the solution of Eq. (2.17”). Then,
Recalling the form of F ; ( z ) , the nature of the changes from x to and from y t o z , and the rules for evaluating the HGlder constant for the product of two functions, we obtain
y(x)
The maximum of the quantity Ia;,(O) - a ; , ( % ) ] is over all z in the support of u k . Let us assume the diameter of the support to be sufficiently small so that
This is the only restriction that we make on the choice of the diameter 6 of the interior regions Q k . For such values of 6, we obtain from (2.20) n
2 I I1kzlZl I(=), i.I=1
En
4 2 c ~I
f
( x ) 12,0, P~ -k 2 c ~1
1,
Q ~ *
(2.22)
For the solutions u k ( z )of Eqs. (2.18”) that satisfy the boundary condition (2.19) (or, more precisely, the condition obtained from it by making the linear transformation from y to z ) , we apply Lemma 2.3. From Eq. (2.14) with condition (2.21) [with constant c taken from (2.14)], we obtain by a procedure analogous to that followed above
(2.23)
I
Tf we use inequality (2.1), we can replace ~ u ~ z i z j ! , z )and ,E,z
“k?.Z.
I ‘k x
~
I1(1).
l2,
01,
En
(Zn”O)
in the left sides of (2.22) and (2.23) by the norms
and 1 U k 12. n, ( Z n ” O ) ’
Returning now to the original variables
and summing these inequalities over all k, we obtain
132
L I N E A R EQUATIONS
I u ( x ) 12. a, 3 -<%(I u 12,0, 3
+I f
In.
3
+I
'p 12,
LI,
s).
(2.24)
From this and from (2.1) we obtain the desired inequality (1.11). It is easy to show that the constant c in (1.11) canbe chosen independently of the dimensions of the region 3. To do this, one should construct the functions Ck ( x ) in such a way that their supports 8, will have diameters not exceeding some fixed number c ( n ) that is independent of the dimensions of 8. Finally, let us prove inequalities (1.12) and (1.13). Let S, be the portion of the boundary S on which the function 'p(s) belongs to C2,a(Sl). A s special cases, S , can coincide with all S or it may even be the empty set. Let u s denote by d , the distance from the point x to S \ S, and let u s denote min ( d x , d y ) by dXy. We introduce the following norms:
We need these norms with 1 = 0, 1, 2. The following relationships are valid for them: lemma 2.4. F o r arbitrary positive E < 1 and an arbitrary func-
tion u ( x ) E c, (a),
MI [ul
< E M , [ill + CE
M , [ u ] ,< E M , + , [ u ]
+
CE
_ _ _I m-'
--
M,,[ u ] , 1
< tn,
(2.25)
l<mr
(2.26)
I
m+'l-l
M o [ u 1.
where the constants c depends on 1, m , and a but not on u ( x ) or E . The proof is elementary but long (cf. [31, 501). We shall not go through it here. We introduce yet one other notation: 2
Let us prove that, for an arbitrary function
N z + ~Q ,Iul,
N2+o,3 Iul
"a.
3
ILnl+ M, Iul
u ( x ) with
+ 1'9 12,
u,
finite norm (2.27)
The constant c in this inequality is determined by the constant of ellipticity v in (1.2), the norms of the coefficients L in Co,II (s), and the norm of the portion S, in C2, The function y ( s ) is u ( x ) IxZs Es,'
..
133
SCHAUDER'S A PRIOR1 E S T I M A T E
Inequality ( 2 . 2 7 ) is a somewhat more precise form of inequality (1.13).
On the basis of ( 2 . 2 6 ) , instead of the norm Nz+a,[ u ] , it will be sufficient for us to find a bound for the quantity M 2 + . , s [ u ] , which, by definition, is equal to
Let us choose two points x and y in Q at which
(2.28)
Two possibilities may arise:
Tn the first case,
Then, inequality ( 2 . 2 7 ) will be a simple consequence of ( 2 . 2 6 ) with I = rn = 2.
In the second case, that is, with I x -yl
d x ,y,
let us
suppose for definiteness that dx,y=d,. Consider the sphere K of radius 3d,/4 with center at the point x. For this sphere, we introduce the function C ( z ) E Cz, (s) defined to be equal to 1 for Ix - z 1 < d x / 2 and equal to 0 for I x - zI >/ 3 d x / 4 . The function v ( z )= u (z)C(z) satisfies the following equation in K n 8 :
where F = Ln . C
+ 2ai~uzlCz,+
aijiiCziz,
+w L i .
Let u s extend w(x) to the entire setG by setting it equal to 0 outside n M. We still use the notation v ( z ) for this extended function.
K
134
LINEAR EQUATIONS
Inequality (l.ll),which we proved above, is applicable to it. For all possible arrangements of K with respect to S,(that i s , whether K does or does not intersect Sl),this inequality canbe written in the following somewhat cruder form:
Let u s assume the function C(z) chosen in such a way that JCt, I
< ..c, d,
lCl(a) < &, and ICzizj( ,< 4 (which, obviously, is possible). After dX
dx
some simple computations, we see that
and
I CY 12, =, K n S , .S 2 +.I 'P 12. , K n S , dx C
If we substitute these inequalities into (2.30), multiply the result by
d?", and keep (2.28) in mind, we obtain
and, on the basis of (2.26), this inequality yields (2.27). This completes the proof of all the assertions made at the beginning of this section for 1 = 2. For 1 > 2, they can be derived in an elementary manner from the case 1 = 2.
3. THE SOLVABILITY IN cz,,(a) OF OTHER BOUNDARY -VALU E PROBLEMS The solvability of the second and third boundary-value problems and the oblique-derivative problem for sufficiently smooth given
THE SOLVABILITY O F OTHER BOUNDARY-VALUE
PROBLEMS
135
conditions is investigated basically in the same way in which this was done in the preceding section for the first boundary-value problem. The analytical basis consists of a priori bounds of the Schauder type, that i s , inequalities of the type (1.11). For the second and third boundary conditions, such inequalities were proven in 1955 by Miranda [51]. These conditions for the operator
take the form Au = a l l ( x ) U X l cos (a. x,)
+b ( x ) u = p ( x ) .
(3.2)
+
For b ( x ) E 0, we have the second boundary condition; for b ( x ) 0, we have the third. However, in contrast with the first boundary condition, we need to impose more stringent restrictions on the functions a , / ( x ) close to S than was necessary for the interior estimates. Specifically, if the function u ( x , belongs to C2,I then it is natural to require that the coefficients in L belong to Co, (G). In this case, every t e r m in Lu, and hence the entire expression L u , will be a function in Co,a(32). The boundary condition (3.2), on the other hand, contains the first derivatives of u ( x ) ; these belong to C,,.(Q). Therefore, the function Q ( X ) must, in the general case, be an element of Cl,m(a)(since, in particular, when the ail are constants and b ( x ) = 0, the function qa is a linear combination of the first derivatives of u ) . But then, we need to require that the a,, be elements of C,,%(S). Thus, there is a difference in the requirements on the a I i ( x ) within 8 and on S. But it is unavoidable. Therefore, when we investigate the second and third boundary-value problems in the nonlinear case, a knowledge only of Miranda's inequalities regarding the linear problems is insufficient. (Miranda found a bound for the norm I u 12, =, for an arbitrary function u in C2, (H)in t e r m s of
(a),
l q n ' m p l ,
and lA~l,,,&s.)
To study the nonlinear problems, a m o r e general result, proven in 1959 by Fiorenza [52] was needed: Theorem 3.1. Suppose that the boundary S of a region 5! i s a class C2. surface, that the coefficients u i i ,a,, and a in the operator L belong to Co,a(G),and that the ellipticity condition
> v z E.: n
a,, ( x ) E,E/
r=1
v = const
>0
(3.3)
is satisfied. Suppose that the coefficients b , ( x ) and b ( x ) in the
136
LINEAR EQUATIONS
boundary operator
are elements of Cl, ( S ) and that n
2 bi (n)cos (n,X i )Is > i= 1
V",
V"
= const
> 0.
(3.4)
Then, f o r an arbitrary finction u ( x ) in C2, (g), (1
and
where the constants c depend only on n, the constants v and v,, in conditions (3.3) and (3,4), and the boundary S , which is assumed to belong to the class C Z s a .
T H E S O L V A B I L I T Y OF OTHER BOUNDARY-VALUE
137
PROBLEMS
The second of these inequalities has the following structure:
where the constant c, is independent of u ( x ) and is determined only by S and the coefficients in the operators L and B . This dependence of c1 on the a l l , the a,, a , the bi, b , and S is clearly shown in (3.5) and (3.6). Knowledge of it is essential for study of the second and third boundary-value problems for quasilinear equations (cf. Chapter 9) but is not significant in the linear case. With the aid of inequality (3.7), we can easily show that, for the problem L u - ~ u = f(X).
(3.8)
Bu(,=~(s),
where h is a complex parameter, the following theorem is valid: Theorem 3.2. Suppose that the conditions of Theovem 3.1
regarding S and the coefficients L and B are satisfied, that f ( x ) E C o .m(%), and that y ( s ) E C,,=(S). Then, .the problem (3.8) has a unique solution in C , , ( B ) for arbitrary f and 'p in the classes mentioned for all h except f o r a countable number of values A,. X,, , . . that constitute the spectrum of the problem (3.8). For these exthe absolute value I l k ( ceptional values k=h,, for k = 1, 2 , approaches infinity (or, more precisely, Re hk + - 00) as k -+00. Each of the kk i s of finite multiplicity, that is, the homogeneous problem corresponding to it
...,
LU -
h k ~
(3.9)
= 0 , BU 1s = 0 ,
has only finitely many linearly independent nontrivial solutions in Cz,a(G). These same h,, for k = 1, 2, , with the same multiplicity, constitute the spectrum of the problem adjoint to (3.9). The inhomogeneous problem (3.8) i s not solvable for all f and 'p; the conditions of orthogonality to the solutions of the homogeneous problem adjoint to (3.9) must be satisfied.
. .,
uk(x)
This theorem is proven in basically the same way as the corresponding assertions were proven for the f i r s t boundary-value problem in Section 1. This time, however, we need continuation along the parameter not only for the operator L but also for the boundary operator B . The overall procedure is a s follows: We denote by M , the operator that assigns to each element v ( x ) in C,,.(a) the pair of functions { L v ; B v ) , which we treat as an element of the space C,,,LI (s) X C!, (S). Let u s consider the family of operators M,v=(L,v; B,v),
where
11,
138
L I N E A R EQUATIONS
and
into For every T in [0, 11, the operator M, maps the space Cz, the space C,,,(n)n Cl, I (S). Inequality (3.7) or, more precisely, the inequality I
I42,01, P
4
c1(
I L u la, P + lB7u 11, a. S)'
which is valid if the coefficient a ( x ) of u satisfies the condition a ( x ) < 0 andb(x) >0, ensures uniform boundedness of the operators M;'(M,-MM,) in the space C z , a ( a )provided we know that M;' exists and is defined on the entire space Co,.@) x C1,"(S). This, together with the fact that the operator M;' is defined for T = 0 on a set that is dense in Co, (n) x Cl, ( S ) , ensures the existence of operators M;' that a r e defined on the entire space C",, bounded (9)x C,, .(Sj for all T i n [0, 11 and, in particular, it ensures a unique solution of the problem (I
BU IS ='p (s)
Lu = f.
(3.10)
in C2,I (G) for arbitrary f in C", ( G ) and 'p in C,,(S). In the general case of arbitrary a ( x ) in Co,=(Q) and b ( x ) in Cl, (S), we have Fredholm's alternative, formulated in Theorem 3.2. The existence of IM,;' on a set that is dense in C, (G) x Cl, (S) can be proven either directly on the basis of the theorems on electrostatic potentials or in the manner in which this was done in Section 1. The inequalities of the type (1.13) with function C ( x ) that a r e necessary for the second procedure hold even in the case that we a r e considering the boundary condtion Bn Is =9. '1
(I
~
4. GENERALIZED SOLUTIONS IN W : ( Q , . THE FIRST FUNDAME NT A L INEQUALlTY
Let us now study equations with unbounded coefficients. Consider equations of the form
under the assumption that the leading coefficients
a,,
a r e bounded
GENERALIZED SOLUTIONS
139
and that Eq. (4.1) is strictly elliptic; that i s , assume that
In Section 2 of Chapter 1,we showed that, to consider generalized solutions of such equations in W: (Q), we need to assume* that
If at least one of these conditions fails to be satisfied, the theory of generalized solutions in W: (Q) is not applicable to such equations since in that case, the theorem on uniqueness “in the small” is violated. We shall say that a function IL ( x , belonging to Wi (Q) is a generalized solution of Eq. (4.1) in W i ( Q ) if it satisfies the integral identity
for an arbitrary function q(x) in W : ( Q ) (cf. Section 1, Chapter 1). It is easy to see that if the functions f iand f and the coefficients in Eq. (4.1) are smooth and if the function u ( x ) is its classical solution or belongs to Wi(Q) and satisfies Eq. (4.1) almost everywhere, then it also satisfies the identity (4.4). To see this, let us multiply (4.1) by q, integrate over 9 , and transform the series of t e r m s by integrating by parts. This leads us to the relation (4.4). The converse is also true: if the function u(x),,belongs to W;(Q) and satisfies the identity (4.4) for arbitrary q E W:(Q) [or at least for arbitrary q E W : (Q)]and if the other functions appearing in (4.4)a r e smooth, then u (x) satisfies Eq. (4.1)almost everywhere. Both these phenomena exemplify the fact that the concept of a generalized solution of Eq. (4.1) is indeed an extension of the old concept of a classical solution. For the identity (4.4) to be meaningful for arbitrary q E W ; (Q), we impose the following restrictions on f l and f:
*For the case q
n a; 3, cf. end of this section.
:
140
LINEAR EQUATIONS
It is easy to verify that, when conditions (4.2), (4.3), and (4.5) a r e satisfied, the integrals of all terms in the identity (4.4) a r e finite for arbitrary u and rl in W : ( Q ) . Therefore, in that case, we may treat q in the definition given above of a generalized solution
a s an arbitrary function in lbl. This broadening of the concept of a solution is convenient from several points of view. In the first place, it makes it possible to study equations of the form (4.1) with nondifferentiable and even discontinuous and unbounded coefficients such that the left side of Eq. (4.1) may not be meaningful. Such equations a r e encountered in a number of problems, for example, in diffraction problems. In the second place, such solutions correspond to one of the principal approaches in functional analysis to the study of linear operators. Specifically, the identity (4.4) defines a bilinear form for the Operator L:
where the elements u a r e to be determined and 7 i:. an arbitrary number of W:(8). A s will be shown in the following section, it is then comparatively simple to determine the u if we also know their boundary values. In the third place, generalized solutions in W: (8) correspond to the physical nature of problems that lead to equations of the form (4.1). We shall clarify this with an example 3f the Dirichlet problem for Poisson’s equation Au== f,
u(,=O.
(4.7)
A s we know, this problem is encountered when we seek the equilibrium position u of an elastic homogeneous membrane that i s fastened along its boundary S and that is subjected to external forces f. The potential energy of such a membrane is given by the integral
I (u) =
J(
IVU12+
2uf)dx.
e
and its equilibrium state, according to Hamilton’s principle, is
GENERAL1 ZED SOLUTIONS
141
determined from the condition that the functional I ( u ) attain its smallest possible value in comparison with all ' ~ in&;@). t From this it follows that, for the equilibrium state u ( x ) , Sl(u)= 2
j-( V u V q + f 7 ) P
d x =0
(4.8)
for arbitrary q in W;(Q). Thus, in accordance with Hamilton's principle, the problem of finding the equilibrium position of the membrane reduces to finding the function u in the class W : (a) that satisfies the integral identity (4.8). The practice that has developed over the years of solving the problem of the minimum for the integral I ( u ) with the aid of the solution to the boundary-value problem for the Euler equation corresponding to it has considerably complicated the problem by introducing an extra requirement not inherent in the physical nature of the problem, namely, the requirement that the solution possess derivatives of twice as high order as the derivatives appearing in the energy integral I ( u ) . In many ways, this departure from the physical situation caused difficulties in solving the variational problems and the Euler equations associated with them. It turns out that the problem of finding the equilibrium position of a membrane is considerably simpler than problem (4.7) for finding the classical solution of the Euler equation corresponding to it. The broadening of the concept of a solution to Eqs. (4.1) proposed above in connection with the problem (4.7) consists in dropping the Euler equation and returning to the identity (4.8), which is simply the integral Eq. (4.4) for Poisson's equation. Let us now make clear what we mean by a generalized solution in W : ( 8 ) of the first boundary-value problem for Eq. (4.1). Obviously, for such solutions u ( x ) to exist, it is necessary that the function 'p (s) giving the boundary values of u ( x ) admits an extension 'p ( x ) to the entire region 8 that belongs to W2 (Q). Thus, suppose that 'p ( x ) E W: (51) and that the boundary condition for the desired solution is (4.9) must belong to W : ( Q ) , it i s sensible to write uls='pls.
Since our solution
u
condition (4.9) in the form u ( x ) -'p ( x ) E lb: (Q). From what has been said, we can see the naturality of the following definition. A generalized solution in W : ( Q ) of the problem(4.1), (4.9) is defined as a function u belonging to W:(Q) that satisfies the identity (4.4) or, what amounts to the same thing, the identity (4.10)
I42
for arbitrary
LINEAR EQUATIONS
(a) and that satisfies the condition
u ( x )-
p ( x ) E &: (Q). Since we shall not be studying generalized solutions in other functional classes in the next two sections, let u s agree for the time being to call them simply generalized solutions of an equation o r of a first boundary-value problem. The boundary-value problem with condition (4.9) is easily reduced to a boundary-value problem with homogeneous boundary conditions. Specifically, instead of u , we introduce the function
(4.11)
w ( x ) = I1 ( x ) - 'p ( x ) .
For this function, we obtain from (4.10) the condition
and from (4.9) the condition (4.13)
v l s =0 ,
which we understand in the sense that w E lb: (a). Instead of the function u , let us seek the function
'u
as a function
in $1 (52) that satisfies the identity (4.12). When we find it, we shall also have the function u =v+'p, which is a generalized solution of the problem (4.1), (4.9). Suppose that conditions (4.2), (4.3), and (4.5) are satisfied. Let u s show that, under these conditions and for ~ ( x in ) W : ( Q ) , the expression
is a linear functional over the space W : ( Q ) and that
(4.15)
GENERAL1Z E D SOLUTIONS
where f = (fl.
..., f,)
and /
and where
143
n
c ( 4 , Q) and c ( 4 , Q) a r e constants in the inequalities
which are valid for arbitrary functions u,E W: (Q) and 7 E ki (8) [cf. formulas (2.19) for E = 1, (2.13), and (2.17 } in Chapter 21. F r o m our assumptions (4.2), (4.3), and (4.5), let us find a bound for Jl(?)(with the aid of inequalities (4.16), Holder's inequality, and the generalized Cauchy inequality [cf. (1.4) and (1.2) in Chapter 21 a s follows:
Inequality (4.15) follows from these inequalities.
I44
L I N E A R EQUATIONS
Let u s turn now to the derivation of the first fundamental inequality (also called the energy inequality) for elliptic operators. We recall that, for an arbitrary function [cf. (2.15), Chapter 21
'u ( x )
E &; (Q),
the inequality
where E is an arbitrary positive number, is satisfied. Let u s represent the coefficient a ( x ) in the form of a difference: a ( x ) = a t ( x ) - a - ( x ) , where a + ( x ) = inax { a ( x ) - a , ; O), a - ( x ) = - a, max [ - a ( x ) + a,; 0) , and
+
I-
d
a ( x )dx.
We define (4.20)
Let u s show that the quadratic form L(w, w ) possesses the following property: l e m m a 4.1. Suppose that conditions (4.2) and (4.3) are satis-
fied, Then, f o r an arbitrary function w E W i (Q),
I [I
VWl2
Q
.++a-+x
4
4 ;(9%v ) f
c , (4)llv 112,,(9)r
where
Proof: By virtue of the ellipticity condition (4.2), we have
1I (v
Q
vw12
+a-wZ) d x Q L (u. w ) + n
(4.21)
GEN ERALl Z E D SOLUTl O N S
145
where E is an arbitrary positive number. Using HGlder's inequality to obtain a bound for the last two t e r m s and using the notation (4.20), we obtain
From this inequality and from (4.22) for
E
= v/2, we obtain
If we bound the last t e r m with the aid of inequality (4.19) and take E=
2M (2v
v24
+ 1)
C*
(4)R '
then, by collecting similar t e r m s , we obtain (4.21). This completes the proof of Lemma 4.1. Let us use inequality (4.21) to find a bound for the generalized solution u of the problem (4.1), (4.9). From the relations (4.12) and (4.13) for the function w = u --'p and from inequality (4.21), we obtain "IVulZ+yo
4
-
4
W q d n ,<,~(~)+~1(4)Il~Il~,ce).
9
where 1(u) is given by Eq. (4.14). (4.15), we have
Then, on the basis of inequality
146
LINEAR EQUATIONS
1
If we transpose the t e r m
I I V V ~ ~ ~t o, ~the ( ~ )left side, collect t e r m s of
like powers, and multiply the result by 2, we get
/
(4.24) 16 c 2 (9. QI 'p* f . vz
f)+
2~*(~~l141;2(Q).
Of special interest are the c a s e s in which we can discard the t e r m ~ ) the right side of this equation. To find out what with l j ~ 1 1 2 ~ (on
these cases a r e , let us use the inequality [cf. (2.14), Chapter 21 I -
(4.25)
Q11vvlI:2(Q) From this it is clear that, if
l l v l l L 2 ( QComes" )~
and the fact that min 51
a- ( x ) = - uo.
2
(4.26)
then it follows from (4.24) that [jVOl2rlX
i
16 <----c2(q. (1 - 6) v2
Q,
'p,
f, f ) .
(4.27)
lf 'p- 0, then v is a solution 11 of Eq. (4.1), and inequalities (4.24) and (4.27) for it take the forms
and
respectively. In the general case for 'pf 0, inequalities (4.24) and (4.27) yield the following inequalities for generalized solutions u = v + ' p of the problem (4.1), (4.9):
I47
GENERAL1 ZED SOLUTIONS
'C&.
P, '9. f. f).
In these inequalities, the constant c ( q , 'p. P, f . f) is takenfrom (4.15) and c l ( q ) from (4.21). Inequality (4.30) is the first fundamental inequality for the solutions of the problem (4.1), (4.9). For ip= 0, instead of this inequality, we can use inequality (4.28); also, when (4.26) is satisfied, we can use inequalities (4.29) and (4.31). We know that inequality (4.26) is satisfied in the follwoing two cases. In the first case, it is satisfied for regions 51 of sufficiently small area since
In the second case, it is satisfied for arbitrary regions provided
(4.32)
It is quite important that this inequality be satisfied for the operators Lu- Au for certain values of 1, for example, for all sufficiently large positive )i. This follows from the fact that the coefficient of u ( x ) in the expression Lu-Au is equal to a ( x ) - ) i and (a ( x ) -A)" = a, - A. In what follows, we shall consider the question of the solvability of the first boundary-value problemfor the entire set of operators L - E with arbitrary complex A. However, it will be convenient to begin this study with values of A such that condition (4.32) is satisfied. Without loss of generality, we may assume that this )i is 0, that is, that inequality (4.32) is satisfied for the operator L itself. The following section is devoted to a study of this case. In Section 2 of Chapter 1, we proved the necessity of the assumption q > n in (4.3) for the propositions proven in the present chapter to be valid. However, in that section, we pointed out that, for n >3, the limiting case q = n retains some of the features of the Dirichlet problem in its classical form. Let u s show this. Suppose that condition (4.2) is satisfied for L and that (4.33)
Let us represent the functions bi ( x ) - a, ( x ) and sums b,(x)-a,(x)=c;(x)+c;(x),
a+ ( x ) in
a+ (x)=c")+c"(x).
the form of (4.34)
148
LINEAR EQUATlONS
where c; ( x ) and c’ ( x ) a r e bounded functions and c; ( x ) and cN ( x ) a r e respectively elements of L,(Q) and L, (Q) with small norms. Suppose that 2
Obviously, in the general case, M., increases without bound a s E’ approaches 0 and the nature of this dependence of Ms.on E’ is determined not only by the size of M in (4.33) but also by L. For L we have an inequality analogous to (4.21). Its derivation i s the same a s for (4.21): We need to findabound for the right-hand member of inequality (4.22) by using Eqs. (4.34). As one can easily see, this leads us to the inequality J’(lvv12+~a-v2)dx,
+ +
e
2 (v 1) --y2-
M 2 r
+
V)+
2(v+
(4.36)
1)
l14!;2(9)yz E’ lrvlr;
2n
n -4y
(Q) *
On the basis of (4.16), the last term on the right does not exceed the quantity
Therefore, if
E‘
i s chosen so that
(4.37) then, we have from (4.36) the desired inequality:
Just a s we derive (4.24) from (4.21), wederive from this inequality the first fundamental inequality for the generalized solutions of Eqs. (4.1) in $;(Q):
SOLVABILITY OF THE F I R S T BOUNDARY-VALUE
PROBLEM
149
In carrying out this derivation, we note that inequality (4.15) is also meaningful for q=n. F r o m this we easily get the corresponding inequality for an aribtrary generalized solution of Eq. (4.1) in W:(Q)that is analogous to inequality (4.30). If the relation I';'
5 [' ;ti --
M,,-rnin 9,
1
2
a - ( x ) cirnes;i-&l
>a,>
0.
(4.40)
holds between the numerical parameters characterizing Eq. (4.1) for Q, c Q , it follows from (4.39) that
for generalized solutions v ( n ) of Eq. (4.1) that belong to $i(Qd. We know that inequality (4.40), like inequality (4.26), is satisfied in the following two cases: (1)for regions Q, of sufficiently small measure and (2) for operators Lu - hu with sufficiently large h >/)lo. However, in the present case, the smallness of m e s '2, and the size of A,, are determined not only by I.( in (4.5) and (4.33) but also b y d and M,. in (4.35) and (4.37). 5. SOLVABILITY OF THE FIRST BOUNDARY-VALUE PROBLEM IN W:(Q) Inequalities (4.29) and (4.41) enable us to a s s e r t the following uniqueness theorem for the problem (4.1), (4.9): Theorem 5.1. The problem (4.1), (4.9) has no move than one generalized solution in W: (9)i f conditions (4,2), (4.3), and (4.26) aye satisfied. For n > 3, these conditions can be replaced by the conditions (4.2), (4.33), and (4.40). Proof: Inequalities (4.29) and (4.41), the right sides of which are zero (since, the inhomogeneous t e r m and the boundary condition are homogeneous), are valid for the difference u = uf -u" of two possible generalized solutions in W i ( Q ) of this problem. Therefore, u ( x ) is equal to 0.
1 50
LINEAR EQUATIONS
It follows from this theorem, in particular, that, for an arbitrary differential operator L that is defined in s1 and that satisfies conditions (4.2) and (4.3), the theorem on the uniqueness of the Dirichlet problem is valid in any sufficiently small (as regards measure) subregion 9,of the region 9,withmes 9,determined only by the constants q, v , and I.* When na3,this is also true for arbitrary L , the coefficients of which satisfy conditions (4.2) and (4.33), but, in this case, mes 8,depends on the choice of L. In the regions Q, c Q of arbitrary size, the uniqueness theorem for the Dirichlet problem is valid for the operators L -hE with h > h,, if t satisfies conditions (4.2) and (4.3) or (4.2) and (4.33) and if ho is sufficiently great. In the first case, h, is determined only by q, v, and p . In the second case, it is determined by the number v in (4.2) and the numbers E' and Me, in (4.37) and (4.40), more precisely, by inequalities (4.37) and the inequality 1-B,
7 -
* Mt,-rninu-(x)-h,, u [Y 0,
1
2 -
cirnes" Q,>8,>0.
Let us now investigate the solvability of the Dirichlet problem for (4.1). We shall do this for the entire set of operators L - E , where )i is an arbitrary complex number, with no restrictions on the smallness of the region 9. Let us suppose that conditions (4.2), (4.3), and (4.5) a r e satisfied for (4.1). Without loss of generality, we may assume that the uniqueness theorem, or, more precisely, condition (4.32), is satisfied for the operator L itself (that is, with A = 0). Then, for a possible solution u of the problem (4.1), (4.9) [throughout this section, we a r e speaking only of generalized solutions in W:(Q)],inequality (4.31) is valid, and, for an arbitrary function 7 ( x ) in
i2(Q),we have
*This is true because
where
I a,, I
mes
--2
'
Q II a II L q , 2 ( Q y
a,= mes-Q
a (x) d x Q
and therefore condition (4.26) i s satisfied for those P the measures of which satisfy the inequality
SOLVABILITY
OF THE FIRST BOUNDARY-VALUE
PROBLEM
151
which follows from (4.21) and (4.32). Let us prove Theorem 5.2. When conditions (4,2), ( 4 3 , and (4.32) a r e satisfied, the problem (4.1), (4.9) has a generalized solution in Wi(S1) f o r arbitrary f(x) in L*; ( Q ) , f(x) i n L,(sZ), and g ( x ) in W:(&).
-
n+2
Proof: We define a new scalar product
in the space $$(sZ). On the basis of our assumptions regarding all and a , the norm 1. I corresponding to this scalar product is equivalent to the norm in the space
fii(L-2).
This is ture because, on the
basis of (4.2) and (4.32), if v c $ $ ( Q ) ,
where
On the other hand, it follows from (4.2) and (4.3) that
from which, on the basis of (4.19), we have
Instead of finding the solution u of the problem (4.1), (4.9), it will be sufficient t o find the function v = u--(g from conditions (4.12) and (4.13). The identity (4.12) can be written in the form
152
LINEAR EQUATIONS
where the expression l ( 7 ) i s defined by Eq. (4.14). Let u s show that the integral
defines, for fixed a in ii(Q), a linear functional on the space $;(Q). That I , ( v , q) is linear inq is obvious; the boundedness of this integral is obvious from inequalities (4.18), which lead to the inequality
and hence, on the basis of (4.19), t o the inequality 111
?)I S~c,(q)ll~ll w;(e)IIT)IIW ; ( Q )
(5.6)
+
with constant c2 (q)= pc (q) I2 c (q)]. According to a theorem of Riesz on linear functionals (cf. [2], p. 396), the functional I,(v, q) can be uniquely represented in the form of the scalar product I , (a.7 )= [Av, ql
(5.7)
Equation (5.7) defines an operator A on an arbitrary element v
G;(Q).
This operator is bounded in and (5.61,
from which we get
Of
i:(Q) since, by virtue of (5.2)
S O L V A B I L I T Y O F THE F I R S T BOUNDARY-VALUE
153
PROBLEM
Let us show that the operator A is completely continuous in i : ( Q ) . Let [ V , ( X ) ) , for m = 1, 2, denote a sequence of elements in the space $i(Q) with uniformly bounded norms
...,
m=l.
2,
... .
...,
Then, the norms of the elements Av,, for m = 1, 2, in @;(Q) a r e also uniformly bounded. Since the operator for the injection of 2n is completely continuous the space Gi(Q) into L,(Q), for p < n--2, (cf. Theorem 2.1, Chapter 2), there are subsequencesof (v,] and [Av,] that converge strongly in the space L,(Q), where p = 24/(4 -2) (we recall that q > n everywhere). Without loss of generality, we may assume that the sequences (v,] and (Av,] themselves converge strongly in L (Q). From the equation
q-2
[ Avi
- Av,,
Avi
-Av,] =1, ( ~ i v,,
Av,
- AV,).
the uniform boundedness of {v,) and ( A v , ) (for m
(5.9)
..
= 1, 2, .) in the norms of W ; ( Q ) and L2q (Q), and from inequality (5.5) a s applied q-2 to the right side of (5.9), it is clear that
Consequently,
(Av,]
converges strongly in the space
i:(Q).
This
proves the complete continuity of A in @:(Q). Under our assumptions on 'p. f , a ndf , the right side of (5.4) is, as was shown in the preceding section [cf. inequality (4.15)] , a known linear functional in $I:@). Therefore, we can represent it as the scalar product of a well-defined element F in W : ( Q ) and r):
L(s)= [ F , sl.
(5.10)
By virtue of (5.7) and (5.10), the identity (5.4) can be written in the form [v+ Av,
4 = [ F , q].
Since this identity must be satisfied for arbitrary 7 in
(5.11)
$/1(Q),
it
154
LINEAR EQUATIONS
follows that (5.11) is equivalent to the operator equation in W i ( Q ) : (5.12)
Av= F.
V+
Since A is a linear completely continuous operator in W i ( Q ) , Fredholm's theorems a r e valid for (5.12). The first of these asserts that Eq. (5.12) has a solution v for arbitrary F in homogeneous equation
$i(Q) if the
(5.13)
w +Aw =O
has only the trivial solution w r O . But any solution of Eq. (5.13) in @:(Q) is nothing other than a generalized solution in W : ( Q ) of the problem (4.1), (4.9) with p = f r f = O since (5.13) is equivalent to the integral identity [w+ Aw, 71 = 0.
and this is simply the identity L(w. ?)=Or
which defines a generalized solution of Eq. (4.1). On the basis of Theorem 5.1, we have w=O, so that Eq. (5.12) does indeed have a solution v for arbitrary F . This solution i s the desired function v since (5.12) i s equivalent to (5.11) and (5.11) is simply the identity (4.12). It determines the solution u of the problem (4.1), (4.9): u ( x ) =v ( x ) p (x). This completes the proof of Theorem 5.2. Now consider the problem
+
Lu-ku=-+
d f1 axi
f.
UIS"'p
(5.14)
for arbitrary complex h. The assumptions on the known quantities in (5.14) remain a s before. In the present section, we shall assume that the Hilbert spaces Lz(Q), W:(Q), and $:(Q) a r e complex-valued spaces and we shall keep the old notations for the scalar products and norms in them. Thus (u. v )
J u ( x ) v ( x )d x .
D
and [ u , v ] denotes
SOLVABILITY OF THE F I R S T BOUNDARY-VALUE
PROBLEM
1.55
In accordance with the definition given in Section 4, we shall refer t o a function u in W&2) that satisfies the identity L(u. i)+h(u.
q)=(f,.
rlx,)-(f.
(5.15)
?)
for arbitrary q in $':(S) with the property that u - Y E i : ( Q ) as a generalized solution of the problem (5.14) in W i ( Q ) . The expression L ( u , 7 ) is the same a s in (4.6). For the function v = u -Q, the identity (5.15) generates the identity (5.16)
L(v. i)+h(v3 7)==4(7)*
where
Just as above, the identity (5.16) can be replaced by an equivalent operator equation in the space c:(S): v
+ AV +hBv
(5.18)
= FA.
where F , is the element in W ; ( Q ) defined by the identity
4 (7) = IF,,
(5.19)
71,
and B is the completely continuous operator in $'A@) identity
defined by the (5.20)
[Bv. T ] = ( v , 7 ) .
It is easy t o see that B is symmetric and positive-definite and hence has an inverse defined on the range R ( B ) of its values. That B is completely continuous is proven in the same way as the complete continuity of A was proven above. A s is shown in Theorem 5.2, the operator E+ A has a bounded inverse in Eq. (5.18) is equivalent to the equation v+~(E+A)-'Bv=(E+A)-~F,.
i;(a).
Therefore, (5.21)
156
LINEAR EQUATIONS
The operator (E+ A)-'& being the product of a bounded operator and a completely continuous one, is completely continuous. Therefore, all of Fredholm's theorems a r e applicable for Eq. (5.21). The homogeneous equation v+A (E
+ A)-'Bv
(5.22)
=0 ,
corresponding to (5.21) is equivalent to the identity
It is natural to refer to the function v in $;(Q) that satisfies (5.23) for arbitrary q in $;(Q) a s a generalized eigenfunction in W:(Q)of the problem Lv =hv,
(5.24)
v l s =0 ,
and to A a s the eigenvalue corresponding to it. According to the second of Fredholm's theorems, all nontrivial (that is, not identically zero) linearly independent solutions of Eqs. (5.22) can be numbered in increasing order of the absolute for k = 1, 2 , , corresponding values of the eigenvalues A=A,, to them. It is easy to show that all the h, a r e located in the interior of some parabola of the form ReA=-cc( ImA(C1,where c and c, a r e determined by the constants n, q, and v in (4.2) and (4.3) with c > 0 and 0 < c, < 1. Let us state these facts in the form of a theorem. Theorem 5.3. Suppose that the conditions of Theorem 5.2 are satisfied. Then, the problem (5.14) has a unique generalized solution in W:(sZ) f o r arbitrary 'Q E Wi(Q),f E L, (Q) and f E L2; (8)
. .,
c
n+l
throughout the complex plane except at a countable set of values A=A, ( f o r k = 1 , 2, ...) such that ~ , I + C O U S k + w . To every A = 1, there corresponds a finite number of linearly independent ) what amounts to the same thing, solutions of Eq, (5.24) in ~ ; ( Q or, of identity (5.23). These exceptional values h,, for k = 1, 2, constitute the spectrum of the problem (5.14). ForA=A,, where k = 1, 2, the problem (5.14) has a solution i f and only i f
...,
...,
lhR( w ~ ) - O ,
where the
w;
I =1,
. . ., N , ,
(5.25)
are all solutions in $;(Q) of the equation (E+ A') w + ~ , B W = 0,
(5.26)
SOLVABILITY
OF T H E F I R S T BOUNDARY-VALUE
PROBLEM
157
that is adjoint t o Eq, (5.22). The number of conditions (5.25) coincides with the number of linearly independent solutions of Eq. (5.22)
for A = A,. We have already proven all the assertions in this theorem except that the conditions for solvability of the problem (5.14) for )i =A, are of the form (5.25). It follows from the third of Fredholm’s theorems that these conditions [for (5.21)l are formulated a s
follows:
[ ( E f A)-’Fi,. z:] = 0,
where the
1 = 1,
. .., N,.
(5.27)
z i are solutions of the equation z +IkB [ ( E
+A)-’].z
= 0,
(5.28)
which i s adjoint to (5.22). Since the operator B has an inverse defined on R ( B ) and since all the k k a r e nonzero, we can, by virtue of (5.19), transform (5.27) to the form
But this requirement coincides with (5.25) if we remember that, on the basis of (5.28), the functions w,6=B-’z: satisfy (5.26). A s one can easily see, Eq. (5.26) is equivalent to the identity pijW.r,Txi Q
+aiwxi7--biWT.ri - awy+xkw<) d x
TE
= 0.
t:(a
which formally corresponds to the equation d L’w -Akw = --(o,,wx -biw) - aiwxi d.Yl
1
+ ow - -Akw = 0.
This completes the proof of Theorem 5.3. It answers the basic question posed at the beginning of the chapter regarding the solvability of the first boundary-value problem for second-order linear elliptic equations. We note that the dependence of f A ( T )on A is caused by the nonhomogeneity of the boundary condition in (5.14). If y r 0 , then l A ( q )= I (T) in (4.15). If, in addition, f zs 0, condition (5.25) denotes ordinary orthogonality o f f to all the w;:
LA, (WL) = I (WL) =-(f. w ; ) = 0.
158
LINEAR EQUATIONS
Theorem 5.4. Suppose that the conditions of Theorem 5.2 are of the form satisfied f o r all the operators L,, for rn = 1, 2, (4.1) by the same constants. Suppose that the sequence {a$ ( x ) } are uniformly bounded and converges almost everywhere to a,] and that the sequences of the functions a?. b y , a", f". f'", and ye"'converge to a,, b,, a , f , f , and 'p in the norms of the spaces L, ( Q ) , L, (Q), Lqn(S), L2(51), L 2; (Q) and Wi(C2) respectively. Then, the sequence of gen__
...,
-
n+2
eralized solutions uNLin W: (Q) of the problems (5.29)
converges strowly in W: ( 8 )to a generalized solution in W: (52) of the limit problem (4.11, (4.9).
The proof is extremely simple and the reader himself can carry it out, using the following outline. On the basis of Theorem 5.2, each of the problems (5.29) has a unique solution u". For vm=pvrn, an identity of the type (4.12) is valid. The same i s true for v = u -'p, where u is a solution of the limiting problem. It follows from these identities that
If we set 7 = v" -v , after several elementary transformations and inequalities of the same type a s in Section 4, we arrive at the desired conclusion that 11 v" -w 11 w ; approaches 0.
Let us consider also the limiting case in which q = n >3. Suppose that the assumptions (4.2), (4.5), and (4.33) a r e satisfied for (4.1) and that, with the aid of some inconsequential simplifications, we have 'p ( x )= 0, that is, that the desired solution u ( x ) vanishes on the contour S. Let us represent the coefficients of the nonleading terms in the forms of sums: ui ( x ) = a; ( x )
+ a;
(X).
+ by ( x ) ,
bi ( x ) = b; ( x )
a ( x ) = a+' ( x )
+ a+" ( x ) .
where a;, b; and a+'' have a small norm in L, (Q), L,, (a), and L,,, (a) respectively and a;. b; and a+' a r e bounded functions, so that
S O L V A B I L I T Y OF T H E F I R S T BOUNDARY-VALUE
PROBLEM
159
Just as above, let us transform the identity (5.15) corresponding to the problem (5.14) t o anequation of the form (5.18), more precisely, to the equation u
+ A‘u $- A”u +ABu = F ,
(5.31)
assuming, without loss of generality, that a - (x) >/ 0. Here, the operator B and the element F are determined by the identities (5.20) and (5.10) respectively and the operators A’ and AN are determined by the identity (5.7) except that we need to replace the functions a,, bi and a+ in I , ( u , 7) with the functions a ; , bj, and a+’ in the case of A’ and with a;. b;, and a+’ in the case of A”. It follows from what we proved above that the operator B is symmetric, positive, definite and completely continuous, that the operator A’ is completely continuous, that A“ is bounded, and that its norm approaches 0 a s P approaches 0. Furthermore, on the basis of Theorem 5.2, the operators E+ A’+A’B have inverses for all sufficiently large A’ (where A‘ >/A;) and (quite important) that the norms of their inverse operators do not exceed some number p2 that depends only on n, v, and 51 and is independent of M,.. This last fact can be seen from inequality (4.39) and the facts that the role of a- in that inequality is now played by aA’ and that 6, can be taken equal to 0 (since, for A’, the functions c; and C” a r e equal to 0). The number Ah, on the other hand, is determined not only by v but also by M,”. Thus,
+
Let us apply the operator (E+ A’+h’B)-’ to both sides of Eq. (5.31) and let u s write the result in the following form: u+(E+
A‘+A”)-’A‘’u$-(h--’)(E+ =( E
A’+A’B)-’Bu
+ A‘ +A’B)-’F.
=
(5.32)
We choose E” sufficiently small so that the norm of the operator A” will be less than l / p 2 . Then, the operator E+ (E+ A’ k’B)-’A’’ will have an inverse, and Eq. (5.32) can be transformed into an equivalent equation of the form
+
u
+ (A - A’)
[E+ ( E
= [ E + (E+ A’
+ A’ +A’B)-’A‘’]-’ + A’B)-’A’’]-’
(E+A’+h’B)-’Bu
(E+ A’
+A‘B)-’F.
=
(5.33)
Tn this equation, the operator multiplied by ( h - h’) is completely continuous (since B is completely continuous and the other operators
1 60
LINEAR EQUATIONS
a r e bounded). Therefore, all the Fredholm theorems a r e applicable to Eq. (5.33). Rewording these theorems in terms of the original Dirichlet problem, we obtain Theorem 5.5. Suppose that conditions (4,2),(4,5), and (4.33) are satisfied f o r Eq. (5.14) and suppose that cp(x)E Wi(Q). Then, all the conclusions of Theorem 5.3 are valid for the problem (5.14). 6. THE SECOND AND THIRD BOUNDARY-VALUE PROBLEMS
The solvability of the second and third boundary-value problems for the equation
in the space W k Q ) is investigated in essentially the same way a s the solvability of the first boundary-value problem was investigated in Sections 4 and 5. We only need to define what it means to solve these problems “properly” in the space W i ( Q ) . For an arbitrary function in W : ( Q ) , it makes no sense to say that it satisfies the condition
du where = a l p x jcos (n. x ~ ) ,where n is the outer normal to S, and dN
where a (s) and cp (s) a r e given functions defined on S, because its first derivatives a r e defined only almost everywhere on an ndimensional region 52 and may fail to be defined (or may be equal to 00) on the entire (n - 1)-dimensional surface S. Therefore, we need to put the condition (6.2) a s well a s Eq. (6.1) in a different form, one that will be suitable for anarbitrary function W i ( Q ) . This can be done a s follows. We shall say that afunction u ( x ) in W i (8) that satisfies the identity
THE SECOND AND THIRD BOUNDARY-VALUE
PROBLEMS
161
zf all the functions in (6.3) were sufficiently smooth, it would not be difficult to verify that when we integrate the first integral in (6.3) by parts, we a r r i v e at the identity
from which, by virtue of the sufficient arbitrariness of 7 , Eqs. (6.1) and (6.2) follow. Thus, the definition that we have given for a generalized solution is indeed an extension of the classical concept of a solution of the problem (6.1), (6.2). Solvability of the problem (6.1), (6.2) in the class k'/:(n) is established in an extremely simple manner. Suppose that the assumptions of Theorem 5.2 a r e satisfied for the coefficients in L and the functions f i and f. Let us first prove the first fundamental inequality. If we set 7 = " in (6.3), we can derive inequality (4.23) just. as was done in Section 4. Then, by using inequalities (2.19) of Chapter 2, we get
where E, is a small positive number and c r , ( a , , b,, a + ) i s a known positive constant that approaches rn a s --t 0. Tf the functions a, (p, and a, were identically equal to 0, the integral over S would vanish and (6.5) would become an inequality of the type (4.24). Here, we do not need to impose any smoothness conditions on S. In the general case, the conditions on a, '9, the a , , and S reduce to the requirement that, for an arbitrary function u in w ~ ( Q ) [-
and
a
+ a, cos (n,
4
xi)] U ? ds I
[E
e
I Fu l2
+
ct(a,
a,) 4 d x
(6.6)
162
LINEAR EQUATIONS
with arbitrarily small
E and finite c,(a, a,) and c(cp) [where the constant c, (a, a,) can increase arbitrarily a s E -+ 01. If the conditions on a, 'p, and the ai a r e formulated in terms of their membership in the spaces L,(S), then, a s the injection theorems show (cf. Section 2, Chapter 2 ) , we need to assume for this that the function a =- a al cos (n, xi)belongs to the space Lp(S) with p > n 1 and '9 EL, (S), where r >/ 2(n - l ) / n when n > 2 and r > 1 when n = 2. We need to assume that the boundary S is piecewise-smooth. When these conditions a r e satisfied, we obtain from the relations (6.5)-(6.7) after a number of simple bound-finding operations analogous to those in Section 4
+
-
This inequality is of the same nature a s inequality (4.24). 'It is used to prove that, for the problem (6.1), (6.2), the three Fredholm theorems hold for the space W : ( Q ) and, in particulr, that, for all sufficiently large )., the boundary-value problem (6.2) for the equations Lu-).u=-+f
dfl
an-,
has a unique solution for arbitrary f, f,, 'p, and a with the properties indicated above. We shall not give proofs of these propositions here since they a r e quite analogous to those given in Section 5 for the first boundary-value problems.
7. INTERIOR ESTIMATES IN L2 OF THE SECOND DERIVATIVES OF A N ARBITRARY FUNCTION IN TERMS OF THE VALUES OF A N ELLIPTIC OPERATOR APPLIED TO IT 'In this and the following section, we shall prove the second fundamental inequality for elliptic operators. 'It enables us to find a bound for the norm in L, ofthe second derivatives of an arbitrary function u in terms of the norms in L, of the function 11 itself and of the values of the elliptic operator applied to it and the norms of the boundary values of u . This inequality was established independently in [55, 10, and 791. The derivation that we give below,
163
INTERIOR ESTIMATES
like all our further investigations on the solvability of boundaryvalue problems in the space Wi(Q),a r e taken from the articles [lo, 11, 121 of one of the present authors. We begin by finding interior bounds that are independent of the values of u on the boundary S. This time, the functions in question belong to the spaces Wi, and the operator L [cf. (4.1)] on them must be calculated directly and must yield functions in L,. To ensure this, in addition to the restrictions (4.2), (4.3), we also require that the generalized derivatives dail/axk and dai/dx, exist , that the dai,/axRbelong to Lq(L2), and that the da,/dx, + a belong to L T n ( 8 ) , where = max (4. 4 ) and q > n.* These conditions ensure that every term in the expression
Lu where a, =%+a, ox,
( x )uXiK/
+ a, ( x ) + i(x)u,
(7.1)
UKi
aai + b, and a = -+ dx,
a
yields, for arbitrary u in
so that L is a bounded operator from W:(Q),a function in b(Q), Wi(Q)into Lz(S) and
II L41'2(Q)scII"Ilw;(Q)
(7.2)
with coeificient c depending only on the constant /.iin (4.2) and the nOrmS II a, II L q ( Y ) and II ^a II L?, (Q). Thus, we write the operator L in the form (7.1) and assume, in addition to conditions (4.2) and (4.3), that
l.1, k = l ,
..., n,
.
q > n , q = m a x ( q ; 4).
Let us first consider the operator L a s applied to functions in that is, to functions Wi(Q)that a r e of compact support in 9. Let us prove the validity of the following proposition: lemma 7.1. If conditions (4,2), (4,3), and (7.3) are satisfied f o r the coefficients of the operator L , then, for an arbitrary function
W:(Q),
u
EWQ),
with constant c depending only on the size of q, ditions (4,2), (4,3), and (7.3) and not on u or&. 'Regarding the possibility of replacing q by n for n situation i s the same here.
v,
and
I-(
in con-
>3, cf. end of Section 4.
The
164
LINEAR EQUATIONS
Proof: The set C, (Q) of all infinitely differentiab!e functions that a r e of compact support in Q is dense in the set W i ( Q ) [in the sense of convergence in W i ( Q ) ] . This is true since the usual averages with indefinitly differentiable kernel and sufficiently small radii of averaging yield an approximation from below for a function in W i (a). Therefore, it will be sufficient to prove inequality (7.4) for functions u inCm(51). For arbitrary u E W i ( Q ) ,this inequality
i s obtained by closure in the norm W i ( Q ) . Here, we need only remember that the operator L, like the operator from W:6?) into L2(Q),is bounded. Thus, suppose that
u f Cm (9). Let
u s consider the integral
+(,^,,,*, + a^u)2]fix. Let u s transform the first term on the right by twice integrating by parts a s follows:
J' (Lu)2 d x = f [SE
3
UX,
d
(a,j a k / " x k x * )
+ . . .] d x =
Let us show that (7.6)
To do this, we choose an arbitrary point xOEQ and set up new rectangular coordinates in a neighborhood of it: yk = a k , ( x l - x;). We choose an orthogonal matrix (an*! in such a way that it will reduce a quadratic form a,,(xO)ErEj to diagonal form, that is, in such a way that
where k1( x ) .
. . . , I,, ( x )
a r e the eigenvalues of the form ail ( x ) ti€,.
165
INTERIOR ESTIMATES
Then, a s one can easily see, n
I, ( x o )=s,2 As ( x o )ht (x") U: s t (xo). t=1 On the basis of the ellipticity condition (4.2) or, what amounts t o the same thing, on the basis of the assumption hi(X)>V,
[=I,
..., n,
we obtain
But
so that inequality (7.6) is established. Therefore, from (7.5), it follows that
The remainder of the proof consists in showing that, for arbitrary E > 0, (7.8)
where the constant c, depends only on E (here, c, 00 as E. --f 0) and the constants q , v , and 1 in conditions (4.3) and (7.3). To prove inequality (7.8), we shall use H61der's inequality, inequality (4.19), and inequality (2.22) of Chapter 2. With the aid of these inequalities, we obtain a bound for the integral I, by the standard procedure. For example, let u s find bounds for certain of the t e r m s in the expression for 12. --f
166
LINEAR EQUATIONS
Let us define
(Here, we are not using a summation convention.) It is easy to see that
To find a bound for the third factor in the second term on the right side of the last inequality, let us use inequality (4.19). Then, we obtain
The bound obtained for the integral I, is an inequality of the type (7.8) since E and a r e arbitrary positive numbers. A s an example, let us consider the expression
for arbitrary values of the indices 1 and j . Obviously,
Let us suppose first that n = 2, 3. Then, by using inequality (2.22) of Chapter 2, we obtain
INTERIOR E S T I M A T E S
167
Suppose now that n >4. Again using inequality (2.22) of Chapter 2, we have
Bounds a r e found 'for the remaining terms in I2 in an analogous manner. Here, in finding a bound for the integral
we need to consider the cases n = 2, 3 and n > 4 separately, just a s we did above. Thus, inequality (7.8) is proven. It follows from (7.7) and (7.8) that
Then, if we take, for example,
E
= +/2, we obtain
This inequality together with inequality (4.24) gives u s the second fundamental inequality (7.4). Let u s draw some conclusions from inequality (7.4). First of all, let us show that the operator L defined above a s the differential operator (7.1) on the set D ( L ) = W ; @ ) has a closure in L2(Q) (cf. [2]). Let { u k ) r Z 1denote a sequence of elements in W i ( Q )that
168
LINEAR EQUATIONS
converges in L, ( 8 )to 0 with the property that [ Lu,} converges in L, (Q) to some element f. We need to show that thenf=O. To do this, note that, from inequality (7.4) a s applied to the function u , -ul, for k , 1 =1,2,...,it follows that the sequence ( u k ]itself converges in the norm of Wi(S1) and, consequently, converges to its limit in the norm of W ; ( 8 ) . But the limit of ( u k ]in L2(&) is u=O. Consequently, ( u k ) converges to 0 in the norm of W i (52) and hence, a s one can easily see, ( L u k )converges to 0; that is, f0. Thus, the operator L admits closure. Furthermore, from what we have just shown, it follows that D ( 0 , that is, the domain of definition of the closure L, coincides with $i(Q), that is, with the closure of the set W ; ( O ) or, what amounts to the same thing, with the closure of C,(U) in the norm of Wi((.?). It is worth noting that on an arbitrary smooth portion of the boundary of the region 9, functions in 1% (8)vanish in mean together with their first derivatives (cf. Section 2, Chapter 2). A s we know, the possibility of closure of the operator L is equivalent to having its adjoint operator L* defined on a dense set in L,(Q). The adjoining operator to L is easily defined in explicit form if', in addition to our other assumptions, we assume that the daij daij that coefficients a,] have derivatives - and dxl,d*ai' that -CL,(P), dx1
dX/
d2aU
d 3 d
EL-(Q), 9
that the
a,
and b, have derivatives 5%
dxi
2
db
and& in
L; (Q), and that the coefficient a belongs to LG (8). Specifically, on
-2
2
'u ( x )
E W i ( Q ) c D (L*), it is defined by
In fact, the conjugate operator L* and its domain of definition D ( L * ) a r e characterized by the fact that, for an arbitrary element u
in D ( L ) = W z ( Q ) , the identity (Lu,u ) = ( u ,
L*V)
(7.11)
holds. But this identity is valid for arbitraryuEWi(Q)if L* is defined by Eq. (7.10) and if v E W:(Q). From all this, it is easy to show that, under certain regularity assumptions regarding S,D ( L ) coincides with W i (Q).
THE SECOND FUNDAMENTAL INEQUALITY
1 69
8. THE SECOND FUNDAMENTAL INEQUALITY FOR ELLIPTIC OPERATORS One of the principal questions associated with an elliptic operator L is that of the solvability of various boundary-value problems for it. These problems consist in finding solutions u ( x ) of the equation
+
Lu = ( x ) (8.1) in a region 9 that satisfy some specified boundary condition on the boundary S of the region 2. The following boundary conditions are considered the basic ones: (1) the first boundary condition
(2) the second boundary condition (8.3) (3) the third boundary condition
where Q (s) and a (s) a r e functions defined on S. From the point of view of the methods that we a r e using, it is convenient to reduce nonhomogeneous boundary conditions (8.2)(8.4) to homogeneous ones. To do this, we replace the unknown function u with a new unknown function z, by setting fJ
(4=2, ( x )
+u1
(x).
where a,(%) belongs to the domain of definition of the differential operator L and satisfies one of the conditions (8.2)-(8.4). Then, v must satisfy the equation Lv =J, -Lu,,
that is, an equation of the same form a s Eq. (8.1) and the corresponding homogeneous boundary condition. Thus, we assume that u satisfies Eq. (8.1) and one of the homogeneous boundary conditions, for example, the condition uJs=0.
(8.2 ')
1 70
LINEAR EQUATl O N S
Trom the point of view of functional analysis, the problem (8.1), (8.2 ) consists in finding the operator L-' inverse to the operator L , yhere L is defined on the set of functions satisfying condition (8.2 ). In the preceding section, we defined the operator L on the set W i ( Q ) . Since functions in W l ( Q ) a r e equal to 0 on the boyndary S, any one of them is a solution of the problem (8.1), (8.2 ) with corresponding right-hand member (namely, q~ = Lu). However, such functions $ [where belongs to R ( L ) , that is, to the range of values of the operator L or even to R ( L ) ] do not fill the entire space L,(M). For example, if we take any twice continuously differentiable 1)
function v ( x ) that vanishes on S such that problem of finding
11
from the conditions Lu=Lv.
uI,=O,
-dn
aw Is
f 0,
and pose the (8.5)
then one of the solutions of this problem is v. Tf we assume that condition (4.26) or condition (4.32) i s satisfied for the operator L. then the function I( = v will be the only possible solution to the
(& L8 (Q) and therefore, problem (8.5). But, obviously, v 4 .!I= Lv @ R Thus, R (L) does not coincide with L, (Q). Thus, if we wish to solve the problem (8.1), (8.2') for all qI E L, ( Q ) or even for all members of a set that is dense in L, (a),we need to extend the operator L. A s we know, even for the simplest elliptic operator, namely, the Laplacian operator, the operator
(z).
on $:(a) has infinite defective and hence it admits an infinite set of distinct extensions. The boundary-value problems for the operator L generate various extensions of the operator L from its original domain of definition D (L)= W i (8)and, in a definite sense, vice versa. Here, we shall consider only one side of this question; specifically, we shall study those extensions of the operator L from W i ( Q ) that a r e associated with the fundamental boundary conditions (8.2)-(8.4). It i s desirable to extend the operator L in such a way that the range of values of the extended operator L^ (or L+AE for some h ) will fill the entire space L,(Q), the functions in the domain of definitionof the operator L^ will satisfy in a certain sense the boundary condition of the boundary-value problem in question, and the operator (or L^+AE) will have a bounded inverse on L,(M). Clearly, for any one of the three fundamental boundary-value problems (8.2)-(8.4), we need to place in the domain of definition D ( i ) of the operator
E,
first of all, all sufficiently smooth functions
THE SECOND FUNDAMENTAL INEQUALITY
171
that satisfy the corresponding homogeneous boundary condition since any such function is a solution of the boundary-value problem in question. For definiteness, let us take the first boundary-value problem. We denote by Wz,o(Q),the closure in the norm of Wi(51) of the set of all functions in C2@) that vanish on the boundary S. Suppose that the region 51 is such that W i , o ( Q ) c ~ ~ ( 5 This 1 ) . will be the case for the classes of regions that we a r e considering, namely, regions with piecewise-smooth boundaries (cf. Section 1, Chapter 1 and Section 2 of Chapter 2). We denote by L^ the extension of the operator L obtained by extending D ( L ) to the set W;,,(Q). A s before, the operator
L^ defined on
W i , , ( O ) is of the form (7.1).
We shall later show that the operator 2 is closed with respect to the set W:,o(Q). Here, we shall show that the operator L^ admits closure and that all the functions in the domain of definition D of the operator ?, have generalized second derivatives that a r e square-summable over an arbitrary strictly interior subregion 52' of the region 8. Both these assertions a r e easily derived from the inequality
(z)
which is valid for arbitrary u in W;(Q). Here, C(x) is any positive indefinitely differentiable function of compact support on the region 51. The constant c(C) in inequality (8.6) depends only on the constant of ellipticity v, the numbers q and p in conditions (4.2), (4.3) and (7.3), and max(K(. Inequality (3.6) is proven in the same way a s inequality (7.4). We only need to consider the expression
instead of
and to transform the highest-order terms a s follows:
172
LI N EAR
--ax, d
( @ l / a k h 2 ) lExkxluxl
E QUATI ONS
$-
a
1d x .
(aijakic2)~ E Xx u x 1 I 1,
On the basis of what was said in Section 7 [cf. (7.6)], the principal positive term
has a lower bound given by
A l l the other terms a r e bounded from above. Most of them differ from the corresponding terms in the expression for
only by the factor C2. Exceptions a r e the terms containing the derivatives of C, namely, the terms
(these disappear for C r 1). We must give a bound for them as follows:
where E is an arbitrary positive number and the constant c depends only on maxlal,l and maxIVC(. lf we go through the remaining reasoning and find the necessary inequalities just exactly a s we did above in deriving the
173
THE S E C O N D F U N D A M E N T A L INEQUALITY
fundamental inequality (7.4), we obtain the inequality
where Q’ is the carrier of C. It is then easy to find a bound for the integral
in t e r m s of
J [ ( L U ) * +u2] d x . P
To do this, consider the integral
J’ Lu . uq2 d x , 8
where ~ ( xis) a smooth function that is equal to unity in to 0 on S , and transform it to the form
L2’
and equal
from which it follows that
The right-hand member of this inequality does not exceed the quantity J[(L1l)2+51VUI~7~+CI(T)U21dX, 8
where c,(ri) is a known constant that depends on
E,
the quantities
174
LINEAR EQUATIONS
max lVql, M , and q in (4.2), (4.3), and (7.3) and where E i s an arbitr a r y positive number. This is shown by means of the same line of reasoning and the same types of bound a s in the first energy inequality (4.24). We only need to apply inequalities (4.19) and (4.16) not to the function u but to the function uq. This leads us to the inequality
which, together with (8.8), yields (8.6). It follows from inequality (8.6) that the operator admits closure (this is proven just a s the assertion that the operator L admits closure was proven in Section 7) and that the functions in D ( z ) will have generalized second derivatives that a r e squaresummable over an arbitrary strictly interior subregion 8' of the region 8 (because, for an arbitrary subregion 0' of this type, we can choose C ( x ) so that C ( x )= 1 when x 6 W). These assertions
a r e valid for all three boundary-value problems (8.2)-(8.4) since inequality (8.6) is valid for an arbitrary function u in Wi(2). $et us turn again to the first boundary-value problem (8.1), (8.2 ) and let us show that, if the boundary S of the region 0 is sufficiently smooth, then D
(z) = W i , (8);that is, the operator
is
closed with respect to W;,,(Q). To do this, we shall extend the second fundamental inequality for elliptic opeators to arbitrary functions in w;. (Q). Let us suppose that the region 52 and its boundary S possess the following properties: (1) a s always, the boundary S is a piecewise-smooth surface with nonzero interior angles (cf. Section 1, Chapter 1); (2) for almost all (in the sense of the measure on S) pointsx" in the surface S , there exists a tangent plane to S and the equation for a portion of the surface S in a neighborhood of the point xu in a local Cartesian coordinate system (with the y,-axis directed along the outer normal to S that passes through xo and the yl-, Y,+~axes lying in the plane tangent to S at the point x o ) is of the form
...,
y, = (Yl.
*
-
9
Y,,-Ib
Here, the function w is twice differentiable and the eigenvalues p, ( x O ) .. . ., p,,-, ( x o ) of the quadratic form
175
THE SECOND F U N D A M E N T A L I N E Q U A L I T Y
at the point xo a r e bounded from above by a nonnegative constant, which we denote by k: (8.10) We shall say that surfaces S possessing these properties are piecewise-smooth surfaces with curvature bounded from below by the number K. For example, for an n-dimensional sphere of radius R, we have pk =- 1 / R for k = 1 , ., n 1. Therefore, for K we may immediately take 0 in the case of all spheres. We note that the principal curvatures of a sphere of radius R a r e equal to 1 / R and thus a r e indeed bounded below by 0. As a second example of such surfaces, we may take the surface of a nondegenerate ndimensional polyhedron (not necessarily convex) or the surface obtained from the surface of such a polyhedron by a topologically twice differentiable transformation with bounded second derivatives and positive jacobian. We shall show now that, if the boundary S of the region Q is a piecewise-smooth surface with curvature bounded below by K , then inequality (7.4) is valid for all functions in C,(8) that vanish on the surface S. lemma 8.1. Suppose that S i s a piecewise-smooth surface with curvature bounded b e l m by a number K. Then, f o r an arbitrary that twice continuously differentiable function u defined on vanishes on S ,
.. -
(8.11)
where the constant c depends only on the constant of ellipticity v of the operator L, on the numbers q and p in conditions (4.2), (4.3), and (7.3)*, and on the surface S (it i s independent both of the function u and the size of the region8). The proof of this lemma is in two parts. The first part coincides with the proof of Lemma 7.1 and reduces to obtaining inequalities (7.6) and (7.8). Specifically, we transform the integral S ( L u ) ?d x D
just a s we did in the proof of Lemma 7.1. Since the function
u
is
*With regard to the possibility of replacing q by n for n > 3, cf. end of Section 4. The situation is the same here.
176
LINEAR EQUATIONS
not of compact support in the region 51 this time, in making the two integrations by parts, we handle the boundary integrals and, therefore, in the right-hand member of Eq. (7.5), we have the following integr a1: /I, d s E S
/
U i j akl[UxkxlUxiCOS(fl, %j)--xlxlUxiCOS(fl.
XR)]~S.
S
Repeating the procedure indicated in the proof of Lemma (7.1), we obtain [cf. inequalities (7.6) and (7.8)]
where E is an arbitrary positive number and c, is a known constant depending on E. We note that Eq. (7.5) with the correction of the boundary integral
and hence inequality (8.12), is valid for anarbitrary function u E C,(Q) although third derivatives of u appeared in an intermediary stage of the derivation. This is true because, by virtue of the piecewise smoothness of the boundary S, an arbitrary function in C2@) can be approximated in the norm of C,@) by functions in the class C,(a) (cf. Section 2, Chapter 2). Therefore, the relationships in which we are interested a r e valid for arbitrary u E C,(Q). Let us now consider Is. Let x3 denote an arbitrary point on the surface S a t which the derivativesdh/dyidyj, for i, j = 1, n 1, exist. Let us take an orthogonal matrix (Ckl) and let us use it to shift from the coordinates ( x l , . . ., x,,) to a local coordinate system
...,
(Yl.
*..)
-
YIJ:
y k = ckl ( x l - xy).
k = 1.
. . ., n ,
(8.13)
where the direction of y,, coincides with the direction of the outer normal at the point x(’. By virtue of the orthogonality of the matrix ( c k l ) , we have x1 - x;=
Ck1Jlk.
1 = 1,
. . .,
n.
(8.14)
THE SECOND FUNDAMENTAL INEQUALITY
177
It follows from (8.14) that cos(n, x l ) = c n 1 ,
Thus, at the point
xo,
1=
1,
. . . , n.
we have
where bpq== a k i C p k C q l . x0
p . 4 = 1,
..
,
n.
Let us now use the boundary condition #Is= 0. Close to the point with coordinates y I = . . . = y , = 0, this condition takes the form u(y,,
...
1
y , l - ] , o(y,.
...,
y,l-J=O.
Close to no, it is identically satisfied in y,. . . ., yn-l. Let US differentiate this identity with respect to y, and y j , for i, j = 1, n 1, remembering that, at the point xlJ,
...,
-
_ do -0, dYl
1=1,
..., n-
1
This yields
at the point xo. With the aid of these relations, we can simplify the expression (8.15) for Is (nu): (8.17) For p = n and arbitrary q or for arbitrary p but q = n, the terms in the square brackets in (8.17) cancel each other out. Therefore, in view of (8.16), Eq. (8.17) takes the form
178
LINEAR EQUATIONS
..
We shall assume that the coordinates y,, . , y n - l in the tangent plane a r e chosen in such a way that all the mixed derivatives a20/aypdy,, for p , q, = 1, ., n 1 vanish at the point x o (which obviously we can always arrange by means of an orthogonal transformation of the coordinates y,, . . ., y n - ] ) . Then,
..
-
(8.19) Therefore, by virtue of property (2) of the surface S and the fact that 0 < bnnbpp - b;,, p2, the inequality
<
where M i s the constant in condition (4.2), is valid for Is(xo). From this inequality and inequality (8.12), it follows that
Since the surface S is piecewise-smooth, it follows [cf. Section 2, Chapter 2, inequality (2.25)] that
where the constant c, depends only on the surface Sand is an arbitrary positive number. When we substitute this into (8.21), we obtain
Then, by choosing E=El=+l
V2
+K(n-
l)p2c1]-',
THE SECOND FUNDAMENTAL INEQUALITY
179
we have
This, together with inequality (2.23) of Chapter 2, yields inequality (8.11). Remark: lt is easy to see that, if 8 is a convex region, then K = 0, I , > O , and the boundary integral
in inequality (8.12) can simply be discarded, and we can derive (8.11) directly from the resulting inequality. In particular, for convex Q and constant a,] that satisfy the condition
where
v
> 0, the inequality (8.22)
is valid for an arbitrary function u ( x ) c Wg.0 (55).
Thus, we have proven the second fundamental inequality for elliptic operators. This was done for an arbitrary function u in the class C,@) that vanishes on the boundary S. But, by definition, such functions are dense in W i , ,,(Q). Therefore, inequality (8.11) is valid for an arbitrary function u in the class W;,o(Q). But then, from inequality (8.11), we get the following theorem: Theorem 8.1. The differential operator i s closed with respect
to the set W i , O(Q). Thus, D ( I )= XI(2)= W:, (Q). It is natural t o expect L^ on W;, o ( Q ) to be that extension of the operator L from the set W i ( Q ) that corresponds t o the first boundary-value problem for the operator L. We a r e interested in those cases in which the second fundamental inequality for the elliptic operator L can be written in the form (8.23)
180
L I N E A R EQUATIONS
that is, without the term IIuIIL2(9)in the right-hand member. This is possible, for example, when the region 8 is sufficiently small or, more generally, when condition (4.26) is satisfied. This is true because, in such a case, inequality (4.29) is satisfied and from it we obtain the following inequality for the function u and the operator
L:
IIUII w ; ( 9 )
IIu II w; (9)4 c IIAfd I1L3(&)'
(8.24)
lf the boundary S of the region !! satisfies the conditions of Lemma 8.1, then, just a s above, we obtain from inequality (8.11) 2
II~ll;;(p) But since u E W:,
< c (IlAullr,(n,+
(8.25)
l I 4 ~2> ( 9 ) ) *
(a),we have
and, on the basis of inequality (4.25), this yields
--2
Let us take E = c02 mes n 52. Then, from inequality (8.25), we obtain inequality (8.24). From Lemma 8.1, we obtain the Corollary. Suppose that uEWi(53) and that the hypotheses of Lemma 8.1 regardirg the region 53 and the coefficients in the operator L are satisfied. Then,
It /I2w; (9)< c [ II Lu II
2:
+ II u /I2, 3 +
(9)
(9)
c1
2
II'9 II w;@)*
(8.26)
where 'Q ( x ) i s an arbitrary function f o r which II ( x ) - '4 ( x ) E W i , o ( Q ) and the constants c and c1 are determined by the same quantities as c in (8.11). Inequality (8.26) is easily derived from inequality (8.11) a s applied to the function u - '9 and from inequality (7.2) for 9.
T H E SECOND F U N D A M E N T A L I N E Q U A L I T Y
181
Zn this section and in the precedingone, we have given two types of inequalities bounding the norms in W i of an arbitrary function u ( x ) : interior bounds (with no assumptions regarding the smoothness of the boundary of S o r the boundary conditions of u ) and bounds that apply to an entire region (in the case of these, assumptions of such a type are necessary). It is also useful to have a bound for the norm of u in W i for a region Q only a portion of the boundary of which satisfies the conditions of Lemma 8.1. For this, we have lemma 8.2. Suppose that a portion S , of the boundary of a of Lemma 8.1 and that C ( x ) is a region 52 satisfies the conditions function that is smooth in 9 , that assumes values in the interval [ 0 , 11, and that vanishes outside S \ S,. Then, f o r an arbitrary ficnction u ( x ) in W: ( 9 )that vanishes on S , ,
(8.27)
Here, the constad c depends only on the quantities v, p , and q in (4.21, (4.31, and (7.3),on max IVCI, and on S , . This Lemma is proven in essentially the same way as Lemma 8.1 and inequality (8.6). Specifically, we take the integral
1C ( L U ) ~
dx
e
and transform it just as in the derivationof inequality (8.11). Since C is nonzero on S,, when we integrate by parts, we isolate the integral
J’ C*uija/zluxi[uxkxl (COS n, x j ) - u x r r j
cos (n, xk)]d ~ .
S,
We can find a bound for it just as we did for the integral
in the proof of Lemma 8.1, remembering that C = 0 on S \ S,. This yields inequality (8.27). Lemmas analogous to Lemmas 8.1 and 8.2 (and even more general inequalities for the derivatives of u of arbitrary 1 > 2 ) are
182
LINEAR EQUATIONS
valid also for other boundary conditions: For conditions (8.3) and (8.4) and for conditions with an “oblique derivative” (in this connection, see [lo-121 of one of the authors). The method explained here is suitable also for these cases. A slight modification is necessary only in the case of the bound for the boundary integral
which is transformed to approximately the same form by using different boundary conditions. Here, the boundary S must belong to c2 Remark: In investigating the convergence of the approximate solutions to boundary-value-problems that are calculated according t o Galerkin’s procedure, the following inequality has proven useful: r .
.
(8.28)
This is a generalization of inequality (8.23) to the case of two distinct elliptic operators. This inequality is valid for an arbitrary function IL in W;, ,,(Q and for any two elliptic operators if the coefficients in these operators satisfy conditions (4.2), (4.3), and (7.3) and if the coefficients a (x) and a (x) in these operators do not, for the function u , exceed negative numbers of sufficiently great absolute value. In the general case, if this last condition is not satisfied, instead of inequality (8.28), we have the inequality (8.29)
These inequalities are proven in the same way as inequalities (8.11) and (8.23). We only need to consider the integral
instead of the integral
J Lu . Lu d x ,
e
transform i t s principal t e r m s by twice integrating by p a r t s just as
ON T H E S O L V A B I L I T Y O F F I R S T BOUNDARY-VALUE
PROBLEM
183
was done above, and use the familiar proposition on the possibility of simultaneously reducing two positve-definite quadratic forms to the sum of squares (cf. 171, 721).
9 . ON THE SOLVABILITY O F THE FIRST BOUNDARY-VALUE PROBLEM IN THE SPACE W;,o(Q) The second fundamental inequality for elliptic operators L enables us to investigate in a comparatively simple manner the solvability of the Dirichlet problem in the space W i ( Q ) . Regarding the boundary of the region Q and the operator i., suppose that the hypotheses under which Lemma 8.1 was proven are satisfied. Let u s consider the following Dirichlet problem in 8: Lu=qJ(x),
nl,=0
(9.1)
f o r q~( x ) E L, (Q). We can assume without loss of generality that the boundary condition is homogeneous. Let us suppose first that the problem (9.1) has no more than one solution in @i(L?); more precisely, let u s suppose to begin with that condition (4.32) is satisfied. In this case, the second fundamental inequality can be written in the form where v is an arbitraryfunction W-i,,,(Q). Onthe other hand, we know [see (7.2)] that Inequalities (9.2) and (9.3) show that the differential operator L s e t s up a one-to-one correspondence between its domain of definition Ws, (8) (we have agreed to denote such an operator by L^) and its range R ( L ) c L&). Let u s show that, when the boundary S possesses a certain degree of regularity, the set H ( L ) coincides with the entire set L,(Q). We shall say that aregionMpossessesproperty :I1 if the problem Au=$,
ul,=O
(9.4)
has a solution in Ws,o(Q) for every set 91 of functions $ ( x ) that is dense in L,(Q). We have the following proposition: Lemma 9.1. Su#pose that the conditions of Lemma 8.1 and inequality (4.32) aye satisfied for L and s. Suppose that 9 possesses
184
L I N E A R EQUATIONS
property $1. Then, the problem (9.1) has a unique solution in W;,,(Q) f o r arbitrary J, ( x ) in L, (8). This lemma is proven in essentially the same way a s Lemma 1.1 of Chapter 3 except that, instead of the correspondence C2,=(G) C,, =@), we now take the correspondence W i , o ( c 2 ) t t L,(Q) and, instead of Schauder's inequality (l.ll),we need to use inequality (9.2). Tn addition, we need to remember that the conditions imposed on L a r e satisfied for the entire set of operators L,u= Au +T ( L -A) u that were introduced in the proof of Lemmas 1.1 and 9.1 and that the constants appearing in these conditions can be taken a s general for all 7 in [0, 11. This method of continuity along a parameter and inequalities (8.11) and (9.3) enable us to assert the validity of the following proposit ion: lemma 9.2. Suppose that the conditions of Lemma 8.1 are satisfied f o r L and S. Then, the operators L,, where 0 ,< T ,< 1, are closed on w ~ , ~ ( S ; )and their ranges L T ( W i , o ( 8 ) )coincide with each other. Just a s in Section 1 of Chapter 3, it is easy to show that condition 'Jz is satisfied for spheres, parallelepipeds, and regions that can be topologically mapped onto a sphere or parallelepiped by a function y = y ( x ) in W i ( & ) , where q > n , with nonzero jacobian. Therefore, from Lemma 9.1, we have Theorem 9.1. Szlppose that the conditions of Lemma 8.1 and inequality (4.32) are satisfied f o r L. Szlppose also that the region 8 either belongs to the class w:, where q > n, o r can be topologically mapped into a parallelepiped by the function y = y (x) in W i (Q), where q > n, with nonzero jacobian. Then, the problem (9.1) has a unique solution in W i ,,(9)f o r arbitrary q~ ( x ) in L, (8). We shall not give a proof of Theorem 9.1 (or Lemma 9.1) here since the proof is completely analogous to the proof of Theorem 1.3 given in Section 1. Instead of inequalities (1.9) and (l.ll), one needs to use inequalities (4.29) and (9.2). Let us say a few words about condition 9. For R ( L ) and L 2 ( Q ) to coincide for the entire set of elliptic operators L with wellbehaved coefficients, this condition is obviously necessary and, if the conditions of Lemma 8.1 and inequality (4.32) a r e satisfied, it is also sufficient (cf. Lemma 9.1). Condition !It is by no means satisfied for all regions. Its satisfaction or nonsatisfaction for regions with angular points, for example, depends on the size of the angles at these points. Let us illustrate this with an example. For 8, we take the circular sector ( 0 -<. r << 1 , 0 ,< 0 0,) in the x , x,-plane. The eigenfunctions of the Laplacian operator for this sector with homogeneous boundary condition have singularities at the point r = 0. For Oo
CONDITIONS
UNDER WHICH GENERALIZED SOLUTIONS
BELONG
I85
singularities are such that the eigenfunctions are elements of Wi(Q), but, for x < I), < 2x, theydonotbelongto Wj(62). Consequently, the solutions of the problem (9.4) in the sector with angle 8, E ( n , 2 ~ ) are not elements of W g , o ( 8 ) for arbitrary + in L , ( Q ) . On the other hand, inequality (8.24) does hold even for such a sector. From all this, it follows that the closure of the operator A from C,(G) leads to the operator A, which is defined on W i , o ( Q )and which maps iVS,o(Q) into the proper subspace L,(8). This operator admits a further extension with preservation of symmetry, and such an extension is necessary if we wish to solve the problem (9.4) for arbitrary ,:t in L,(Q). It is easy to show that this extension is unique and is obtained by adding to the set W i , o ( Y )all elements of the n
form crKj;f(r)sis --, where xo
00
c
is an arbitrary constant and f ( r ) is
any twice continuously differentiable function that is equal to 1 close to r = 0. On the other hand, if fJ,& r , then A (and hence any other symmetric elliptic operator L ) is self-adjoint even on the set W i , o ( Q ) . ln the following sections, we shall show that the differentiability properties of the solutions of elliptic equations are local properties, that is, that they depend on the corresponding characteristics L ,+ , and S only in aneighborhood of the point in question. By using this fact together with an example of Guseva and inequalities (8.11), (8.27), and (8.28), and also by using certain facts in the theory of the extension of symmetric operators, Birman and Skvortsov have shown [53] that the defective number of the operator I. is exactly the same as the number of angles exceeding n. This makes it possible to describe all selfadjoint extensions of the operator L.
10. CONDITIONS UNDER WHICH GENERALIZED SOLUTIONS IN W:(S)BELONG TO Wi(S2) Suppose that the coefficients of the operator
satisfy conditions (4.2) and (4.3) and suppose that u is a generalized solution in Wi(i2) of the equation
for
fl
E L, (8) and f E L *; (a).
-
n+2
186
LINEAR EQUATIONS
We do not make any assumptions of the type of inequalities (4.32) or, more generally, any assumptions a s to whether h = 0 i s or is not a point of the spectrum for L under some boundary condition. Thus, in particular, II can be an eigenfunction for I , under some boundary condition. The purpose of the present section is to prove that, if in some subregion Q' of the region Q , the coeffieients of the operator L also satisfy conditions (7.3) and if the function (i, belongs to Lz(Q'), then the solution u has generalized .second derivatives in that subregion that are square-summable over &''c Q' and it satisfies Eq. (10.1) for almost all x in M'. On the other hand, if Q' is adjacent to a sufficiently smooth portion S, of the boundary of s1 and if the values of u on that portion coincide with the values of some function Q ( X ) € Wg(Q),then u ( x ) will have square-summable second derivatives close to S , also. In particular, if 8' coincides with Q , if the boundary S possesses a specified degree of smoothness and if the generalized solution u of Eq. (10.1) in the class W : ( Q ) coincides on S with the function Q ( x ) Wg(Q), ~ then u must belong to W:(Q). Let us suppose first that 51' i s an interior subregion of Q . Let u s take an arbitrary sphere K, that belongs to 8' and that has a radius p sufficiently small for condition (4.26) to be satisfied for the operator L in K,, so that
4-ai??xi - b,v7xl - a?2)d x
Lp (71 7)
(aij7xi?x, KP
(10.2)
zz cllsllf;(,P)
with constant c > 0 for arbitrary 7 E W i (K,). Let us construct a sequence of infinitely differentiable functions u,(x), for m = 1, 2, that converges to the solution II in the norm of W: (K,) and let us consider the boundary-value problems
...,
Lv=+,
m=1,
2. ...,
(10.3)
in the sphere K,, where S, is the boundary of the sphere K;. On the basis of Theorem 9.1, each such problem has a unique solution v, in the space W:(K ). Inequality (4.31) holds for the functions w m ; also, the right side of this inequality is uniformly bounded for all tri = 1 , 2, that is, l / v m ~ l w ; ( K p ) 4m c ,= l .
2.
... .
...;
(10.4)
Furthermore, inequality (8.6) is also valid for w,,, and its right side is, on the basis of (10.4), also uniformly bounded for all m = 1, 2, that is,
...;
(10.5)
CONDITIONS UNDER WHICH G E N E R A L I Z E D SOLUTIONS BELONG
187
Here, C (x) is an arbitrary twice continuously differentiable nonnegative function that is equal t o 0 close to the boundary S,. A s a consequence of (10.4) and (10.5), we can extract from the sequence [ v m ]a subsequence ( v , ~that ~ ~ converges ) weakly in the norms
i 4with arbitrary
of W:(K,) and W i K v 'u
T
= 2, 3,
...to some function
[I.
that also satisfies inequalities (10.4) and (10.5). Let u s show that the equations coincides with For the functions u and
a r e valid for arbitrary 7 in Wi(K,). Therefore, L, (u - V / f I k . T ) = 0.
Let u s now set 7 = umk-v , , ~ . Then, L, (urnk -'Urnk.
u7ik
- vm,)
+L, (u -urnk. umk
Therefore, in view of The second t e r m approaches 0 as k -+a. + 0 also. This shows that ii =v in K,. (10.2)s /I u,nk - vmk "w:(Kp) Thus, for interior regions Q', the assertion made at the beginning of this section is proven. Suppose nowthat the subregion M, is adjacent t o the boundary of the region s1. We may assume without loss of generality that 51, is sufficiently small that inequality (4.26), and hence inequality (4.31), is satisfied for the operator L in Q,. Furthermore, let u s suppose that 8, satisfies the conditions imposed on the region in Theorem 9.1. This assumption imposes a restriction on that portion S, of the boundary S that is common to the boundaries of 52 and C,. Let u s assume that u I s , = 0 (otherwise, we would only need t o subtract from u ( x ) a function p ( x ) that coincides on S, with IL and belongs to Wz ( 2 , )and then c a r r y out our calculations for the function 11 (x) - 9 (x)). We introduce a sequence of functions [I,, for m = 1, 2, that vanish on S,, that belong to W ; ( Q , ) , and that converge to u in the norm of Wi(8,).Let u s consider the problem
...,
L"J= y' .
vls,=u,,,Is"
m=1.
2,
....
(10.6)
in the region Q , , where S' is the boundary of B,. On the basis Of Theorem 9.1, the problem (10.6) has a unique solution v, in the class Wi(Ql). On the basis of inequalities (4.31) and (8.27), (10.7)
188
L I N E A R EQUATIONS
with constants independent of m. Here, C(x) is an arbitrary nonnegative twice continuously differentiable function that vanishes close to S'\S,, that is, t o that part of the boundary S' that does not belong to S. Then, just a s above, we conclude that the function representing the limit of the sequence (71,) satisfies inequalities (10.7) and coincides in 8, with the solution u. In this way, we investigate the differentiability properties of u close to the boundary. Together with the results of the investigation of u within P, this enables us to draw conclusions a s to whetherthe solution u belongs t o W i ( Q ) . Let u s summarize all this in the form of Theorem 10.1. Let U ( X ) be a generalized solution in W:(P)of Eq. (10.1). Let us suppose that the coefficients and inhomogeneous terms in (10.1) satisfy the conditions n
(10.8)
Suppose that, for some interior subregion 8' of the region 9, the coefficients of the operator L satisfy not only conditions (10.8) but also the conditions* (10.9)
where
4rmax (7,
4) and
(10.10)
Then, the solution u (x) has generalized second derivatives in 9;it satisfies Eq, (10.1) for almost all x in 9', and
Ill41 w g ( o , ) Q C ( V 9
P I
4.
~)p4f2(Qr)+ ll$Ilf2(Q,)]*
(10.11)
where C is a smooth nonnegative function of compact support on s?'. 'With regard to the permissibility of replacing The situation i s the same here.
11
by
II
for n
>3, cf. end of Section 4.
C O N D I T I O N S UNDER W H I C H G E N E R A L I Z E D SOLUTIONS BELONG
189
Suppose that the boundary of a region P, c 0 coincides in part with the boundary S of the region Q. Let us denote this common portion by S,. Suppose that 8 , satisfies the conditions imposed on ihe region in Theorem 9.1. Suppose also that the bounday, values of u on S , are given by a function q ( x ) belonging to W:(2,). Then, the assertions just made remain valid for the solution u in the subregion 9, in a strengthened f o r m inasmuch as the ficnction C ( x ) in (10.11) i s necessarily equal to 0 only in a neighborhood of that Portionof the boundary - of- 52,. that is not contained in S. (In this case, we need- to add Ily(lw 2 I to the expression in the square brackets in ( l O . l l ) . ) 2 ( I! Swpose that conditions (10.9) and (10.10) are valid f o r 8 , = % that Y i s a region of the type indicated in Theorem 9.1, and that ( x ) defining the boundary values of u belongs to W ; (9). Then, u E W i ( Q ) and (10.12)
For such regions 8 ,
The proof of this theorem follows from the considerations made above if we note that regions satisfying the conditions of Theorem 10.1 can be partitioned into regions of the same type but of sufficiently small size. Such a supplementary partition of 8, or 2.! in the general case must be performed because, in proving that u belongs t o W; @,), we assumed above that the size of 8 , was sufficiently small so that inequality (4.26) is satisfied for L in 8,. The restrictions that must be imposed on the coefficients in the operator L and on the function II) in order for an arbitrary generalized solution u of Eq. (10.1) to belong to W i ( Q ) are due to the nature of the problem. They are such that every t e r m in the expression for Lu, when written in the form (10.1) and in the form
for u W ; ( Q ) , belongs to L2(S2) and all the t e r m s other than the principal t e r m al j U x , r , determining the type of equation are subject to a l p x i x , in the sense that, for these terms,
1 90
LINEAR EO'JATIONS
with arbitrarily small positive E and with. c, increasing without bound a s E-+ 0. This requirement i s expressed in terms of the membership in the different sets L,(Q) of the coefficients of the unknown function u in the expression L , and in these terms it cannot be weakened. 11. OTHER WAYS OF PROVING THE SECOND FUNDAMENTAL INEQUALITY
Zn Sections 7 and 8, we derived the second fundamental inequality for elliptic operators of the form
For such operators, the requirement of differentiability of the coefficients a , , and u i i s caused by thenature of the situation. However, if the operator is of the form (11.1)
it is natural to seek to prove the second fundamental inequality (11.2)
for u E Wi, (51) without assuming differentiability of the coefficients in M. One might think that a sufficient assumption, in addition to the assumption of ellipticity, would be boundedness of maxJa,j(and of the norms
8
However, example (2.19) of Chapter 1 shows that, for n > 2 , boundedness of the u i j is insufficient. Let us show that inequality (11.2) holds for continuous ail ( x ) with the constant c depending on the modulus of continuity of the uti. We shall not give a complete proof of this proposition. The basic idea involved i s the same a s Schauder's idea for proving inequality (l.ll),which gives a
191
O T H E R W A Y S OF P R O V I N G T H E S E C O N D F U N D A M E N T A L I N E Q U A L I T Y
bound for 1u12,a,Q.* It consists in reducing the entire question to finding the corresponding bounds in small regions for elliptic operators in which the leading coefficients a i j a r e replaced by their values at some point in the small region chosen and to combining these bounds by using the continuity of thenij(x). ln connection with this idea, the general outline of the proof of inequality (11.2) is as follows: Suppose that II E IV;. (9). We denote a value of the operator M applied to u by + ( x ) . Let u s cover the region Q with small overlapping regions Q k c 8 , where k = 1, N , the boundaries of which a r e piecewise-smooth surfaces with curvature bounded below by a number K (cf. definitions in Section 8). By virtue of the continuity of the coefficients their values at points of the region52, differ only slightly from their values at some fixed point x t of that region. We write the equation Mu = 9 in the form
...,
M,u
=a,
(xi) uxi
i
- qJ - [ a i ( x ) - a, (xo")]U X i X j - niuxi - au = F -
.
We recall that the coefficient of uXixj on the right is sufficiently small. F o r operators Mo in Q,, we have inequalities of the type (8.27), specifically, llCu II w;( Q k )
s
c1
I1 Mou II L2 ( Q k )
+
CPll u
/I w; (Bk)
(11.3)
with constant c1 depending only on the constant v in the ellipticity conditions (4.2). The right-hand member of (11.3) does not exceed the quantity 1
t C 3
( ll~ll,;(,k)+
II+IIL2(sk)).
*In a footnote in [ 9 I], this idea is described by Schauder in connection with the derivation of inequality (11.2) for the c a s e of plane regions (i,e., n -1 2) and continuous coefficients ail. Schauder knew that such inequalities had been proven e a r l i e r by Bernstein in [221 (see also [54]) for equations of the form 2
in a c i r c l e with a r b i t r a r y measurable coefficients ail ( x ) satisfying the inequalities 2
L
Y
2 El i=l
al
p
2 ET i=1
and for general equations of the form
Mu = f with differen-
tiable coefficients aij ( x ) , ai ( x ) and a ( x ) , where a ( x ) 0 in convex regions. Schauder himself felt that his principal achievement was removing Rernstein's assumption that the region be convex. These r e s u l t s a r e omitted in Soviet literature up to the middle fifties and they do not appear in the surveys of Bernstein and Petrovskiy.
192
L I N E A R EQUATI ONS
...,
Let u s sum (11.3) over all regions Q k , for k = 1, N , remembering that the function L can be chosen nonzero on that portion of the boundary 3f 9, belonging to S (we assume that u vanishes on S). Using the inequalities that we have, we obtain the inequality
If we choose the regions Q, so small that c 4 m a x n i a x ( ~ , ~ ( x ) - - ~ ~ ( X4 ~ )2 I1 , i, j. k x E Q k
(11.5)
we obtain from (11.4) (11.6) Again using the inequality (11.7) with arbitrary E > 0, we obtain inequality (11.2) from inequality (11.6). From these remarks, the reader himself will be able to c a r r y out the complete proof of inequality (11.2). He need only note that in inequality (11.3) the constant c, can be chosen independently of the region Q k and that it is determined only by the smallest eigenvalue v of the quadratic form aljS,Ej. This is clear from the derivation of inequalities (7.4), (8.20), (8.22), and (8.27). Specifically, inequality (7.6) gives u s a lower bound for the principal positive term; bounds for all the other t e r m s are obtained in t e r m s of it and the derivatives of u of lower order. Let u s state the result that we have been describing in the form of l e m m a 11 .l. Suppose that the regionP satisfies the same conditions as in Lemma 8.1. If the coefficients a,! in the operator M defined by the expression (11.1) are continuous in C and i f the ( ~ ) , llailL ;(Q), where q^=max [ q , 41 f o r q > n , are n o m s I ~ U , J I ~ ~ and -
finite, then, f o r an arbitr&'jy function u in W i , o ( Q ) , inequality (11.2) is valid with constant c depending on the region 51, the values of v and u in the double inequality
193
OTHER WAYS O F PROVING THE SECOND FUNDAMENTAL INEQUALITY
n
n
the modulus of continuity ofa,,,undthe norms IlalllLo(Q)and Ilall,
T(n).
-_
A s will be shown in Section 17, for n = 2 we can discard the 'requirement that the a,,(") be continuous. For n >2, we cannot do this. Keeping the scheme for proving inequality (11.2) that we have been discussing, one can show that, for elliptic operators M, the following more general inequality is valid:
(11.8) where p is an arbitrary number greater than 1 and u ( x ) is an arbitrary function in Wyl,o(Q). With regard to thecoefficients in M, we need to assume that the a,, ( x ) are continuous and satisfy inequalities (4.2), that the i,( x ) are q-summable over Sq (where q > n) if p < n and p-summable if p > n , and that a ( x ) is (q/2)-summable (where q > n ) over 2 if p ,< n and (p/2)-summable if p > n. The constant c,,,, is determined by n, r , p , v, ,u, the modulus of continuity of the a l l @ ) , and the norms of a , and a in the spaces L , ( Q ) indicated. It also depends on the boundary S, which is assumed to be twice continuously differentiable (or at least belongs to W i , where q = max ( p ; n + E ) for E > 0). Proofs of inequality (11.8) for the case of bounded coefficients a, and a appear in the articles [86] and [87].
12. CONDITIONS UNDER WHICH GENERALIZED SOLUTIONS IN Wz BELONG TO Cl,a FOR C 2
>
Suppose that the function u ( x ) belongs to %It, and for almost all
x in Q , satisfies the equation Lu =a,/ ( x ) u x p j
+
a , (4u x ,
+a
( x ) =f
(
4
1
(12.1)
and inhomogeneous t e r m in which are are elements the coefficients of C,+ (Q), where 1 2. Suppose that
>
where Y > 0. Here, !Dl is the set of elements Wi(S2) with finite essential max. Let u s show that u ( x ) actually belongs to Cl,a[Q)a Q
We need only do this for an arbitrary sphere K, of small radius p.
1 94
LINEAR EQUATIONS
We choose the size of the radius p in accordance with the coefficients in L as follows: F i r s t , we take p sufficiently small that, for the operators Lou E a i f (x ) uXix,+ ai ( x ) uxi 0
+a ( x ) u , where x" E K,.
and an arbitrary function v ( x ) in W!,o(K,), the inequality (12.2) is satisfied.
In the second place, it is necessary that inequality (1.9) be satisfied for L" in K , and hence that the theorem on the uniqueness in the class C,, =(Kp)be valid. Finally, we need to take p such that cn
rnax ( a i j ( x f ' ) - a i , ( x ) l i. 1: xE Kp
.< -,21
(12.3)
where c is the constant in inequality (12.2). Obviously, we can satisfy all three of these requirements if we take p sufficiently small. of infinitely difLet u s take a sequence ( u r n ) ,for rn = 1, 2, ferentiable functions in the sphere K , that is uniformly bounded and that converges in the norm of W:(K,) to the function u (x). For each of the u,,~,let u s consider in Upthe problem
...,
Lvm = f *
Vmlsp
=~rnl~~*
(12.4)
where S, is the boundary of Up. From what was said above, the problem (12.4) has a unique solution v, in C l , (K,). The difference wnl= u, -v, satisfies the equation Lwn1=Lu, -j, which we write in the form Low, = (L" - t)w,,
+ Lu,
-f.
(12.5)
and the homogeneous boundary condition wrnlSp=O.
For this difference, inequality (12.2) yields
But (Lo- L) w, = [a,, ( x o )- ui/ (x)]w , , , ~ ~and, ~ ] , therefore,
From this inequality and inequality (12.6), we have by virtue of the
I95
T H E BOUNDEDNESS O F G E N E R A L I Z E D SOLUTIONS
assumption (12.3) ,
11% I1 w; (Kp) ,< 2c IILu, -f II Lq (K,)’
(12.7)
A s one can easily see, the right side of this inequality approaches zero a s m - m , so that {vm) converges in W i ( K , ) to u. But for wm, where rn = 1, 2, we have the uniform bounds with respect to rn [cf. (1.13) and (1.9)] IemLIi, a, Kp 4 c (1, a, C) ( I fie, K, max Iv m 1). (12.8)
...,
+
K?
where C ( x ) is any function in Cl, (K,) that is of compact support in K,. The constant c ( 1 , a. L) depends on this function and on 1 and a but is independent of m. It follows from (12.8) that the function u representing the limit of the sequence (vm)will belong to C,, ( K , \ s,) and will satisfy inequalities (12.8). The desired assertion is proven. Analogously, by using inequalities (1.11) and (1.12) instead of (12.8), we can show that u ( x ) belongs to Cl. (G)o r to Cl, (S2 US,) if we also assume that the boundary values of u on S (resp. on the portion S , of the boundary S ) belong to C l , aand if S itself (resp. S,) is a class Cl,a surface. Slight additions to our reasoning that a r e necessary for this are described at the end of Section 10. Let u s formulate these assertions in the form of Theorem 12.1. S q p o s e that u ( x ) E ID{ and satisfies almost everywhere in Q the elliptic equation (12.1), the coefficients and inhomogeneous term, in which are functions in C , - 2 , u ( E ) ,where l>, 2. Then, u ( x ) i s a function in C,p ( Q ) . If, in addition, the boundary S (resp. the portion of it S , ) i s a surface in the class C l , gand if the boundary values of u on S (resp. on S , ) are defined by a function in the class Cl, a, then u ( x ) belongs to C l , , [resp. to C,, (Q u SJ]. (L
‘I
11
(a)
~
13. THE BOUNDEDNESS OF GENERALIZED SOLUTIONS IN Wi(Q) Suppose that u ( x ) is a generalized solution in W:( Q ) of the equation (13.1)
and that the coefficients L satisfy the conditions*
(13.2) 2
‘For the c a s e 4 = n
> 3, cf. end of this section.
196
LINEAR EQUATIONS
In Sections 4 and 5, we established theorems on the solvability in W: (Q) of the Dirichlet problem for (13.1) under the assumptions that f E L2;i( Q ) ,that f rE L, (M), and that the boundary values of u are given ;+z
by a function ‘9 ( x ) E Wi (52). In this and in the following section, we shall show that under somewhat better properties of f and the f i , an arbitrary solution of this type belongs t o the Halder class Co,m(’sz). In Section 2 of Chapter 1 , we showed with the examples presented there that the following conditions a r e necessary for this: (13.3)
Let u s show that these are also sufficient conditions for u to belong to C,,(Q). Furthermore, we shall obtain bounds for the quantities max IuI and I u I , in t e r m s of the constants Y and 1 in conditions (13.2) and (13.3) and for I I u I I ~ , ( ~ ) . Of course, the bounds for these norms for all Q depend also on the corresponding characteristics of the behavior of the function u on S. Thus, let u s suppose that (13.2) and (13.3) a r e satisfied and that u is a generalized solution of Eq. (13.1) in W:(Q), that is, that u belongs to W : ( Q ) and satisfies the identity
for an arbitrary function 7 E &: (Q). Now, let u s take for 7 the function ~(x)=C*(x)max ( u ( x ) - k ; 01,
where k is an arbitrary positive number and C(x) is an arbitrary smooth nonnegative function of compact support on an arbitrary sphere K , c Q whose values lie between 0 and 1. Such an ~ ( x is) an admissible function since, on the basis of Lemma 3.3 of Chapter 2 , it belongs to the class lb:(Q).The function q is nonzero only on the set dk, of points x in K , , where u ( x ) > k. If we substitute this function into (13.4), we obtain after some simple calculations,
J ( a i i l f x i u1x C * + a i j ~ x , ( u - - ) . ‘k, p
+(up
-ffl)[Ux,C2+
XC~,+
2 ~ c x i ( u - k ) ] - ( ~ ~ u x ~ + a ux- ~
x ( u - k ) CZ} d x = 0.
(13.5)
197
T H E BOUNDEDNESS O F G E N E R A L I Z E D SOLUTIONS
On the basis of (13.2) the first t e r m is not less than v l V u ) 2 [ 2 . Let u s leave this t e r m on the left and find a bound for it from below. Let u s transpose the other t e r m s in (13.5) to the right and bound them from above, using inequalities (1.1) and (1.2) of Chapter 2 giving a small E to factors containing nXi. This leads to the inequality
(13.6)
Let us now set
E
= ~/6.Then,
(13.7)
From this, we get the inequality
(13.8)
where
and c is a constant depending only on v and 1 in (13.2). On the basis of the assumptions (13.2) and (13.3), the quantity l(DII,q(9)is finite and is determined only by the size of
p.
2
I98
LINEAR EQUATIONS
Let u s bound the last integral in (13.8) by using Halder's inequality:
where c, is determined only by v, p , and q in (13.2) and (13.3). By virtue of inequality (2.13) of Chapter 2, the first t e r m on the right in inequality (13.10) satisfies the inequality
-
= 2/n 2/q > 0 and the constant c p ( q )depends only on n where and q. If we assume the radius p is such that c2 (4) c1cp"eI
1
4y
(13.12)
I
then, on the basis of (13.10) and (13.11), it follows from (13.8) that JIVu12C2dx p
q
.f(u-kh)2IVCl~dn+
'k0
(13.13)
p
+(k2+
1)mcs
I--;
:A,,,].
The constant 7 is determined only by the quantities v , p , and y. Inequality (13.13) is proven for all k and p that satisfy condition (13.12). Let us set C equal to unity within the sphere KP-.IP,concentric with K,,where a is anarbitrarynumber in (0, 1) and otherwise choose C so that it satisfies the condition I V C J ,.< c/op. Then, inequalities (5.12) of Chapter 2 follow from (13.13), so that we conclude from Theorem 5.3 of Chapter 2 that, for arbitrary Q ' c Q , the quantity esssential max u is finite and is bounded above by an ?'
expression involving only 7, q , I/ u 11 L 2 ( Q ) , and the distance between !!' and S.
199
T H E B O U N D E D N E S S OF G E N E R A L I Z E D S O L U T I O N S
Analogous considerations for the function - u ( x ) lead to the bound from below for essential min u. Thus, we have bounds from P'
above and below for the expression essential rnaxlul. Q'
Inequalities (13.13) remain valid for spheres K, that intersect S provided the k satisfy the inequality k
> inax [ fu ( x ) ] . Kpns
Tn accordance with Theorem 5.3 of Chapter 2, this enables us to give a bound for essential max [ + u ( x ) ] close to those portions S , of the boundary S on which the quantity essential max [ k: u ( x ) ] is S,
finite. In particular, if M, = essential max/u (x)l is finite, we are in s
a position to give a bound for essential maxlu(x)l. We can do this 9
last not only by using Theorem 5.3 of Chapter 2 but also by using the simpler Theorem 5.1 of Chapter 2. .Specifically, if M,, < 00, then we set 7 (x)= inax ( u ( x )- k ; 01, where k > M,,, in (13.4). Such an After setting up a sequence of inequalities analogous to those above (these are somewhat simpler since all t e r m s containing derivatives of the function C, which in the present case is identically equal t o 1, drop out), we obtain the inequalities ~ ( x belongs ) to $:(Q).
with constant 7 depending only on q, v, and p . Here, k is an a r bitrary number greater thanM,and A, is the set of points x in 9 at which u ( x ) > k. Analogous inequalities can be established for the function - u ( x ) . On the basis of Theorem 5.1 of Chapter 2 , these inequalities together imply the boundedness of essential max Iu ( x ) l 2
and the possibility of giving a bound for this quantity in t e r m s of v. P. 4. and lluilL,cp,* Thus, we have proven Theorem 13.1. Suppose that u ( x ) i s a generalized solution in W: (9) of Eq, (13.1) and suppose that conditions (13.2) and (13.3) are , quantity essential max satisfied. Then, for an arbitrary V c Q the P'
( u ( x ) l is finite and on v, p q, (1 u /IL1(*,,
bounded from above by a constant depending only , and the distance from Q' to the boundary S of the region 9. If, in addition, essential max 1uix)l < 03 f o r some portion S,
S , of the boundary S , then essential maxlu(x)J,where Q, i s a sub%
region of Q that lies at a positive distance from S \ S , , is finite and
200
L I N E A R EQUATIONS
bounded f r o m above by a constant depending only on v, p , q, /Iu 11 L,(p,, and the distance from 51, to S \ S , . In particular, i f essential max lu(x)l
< 00, then
s
essential max I U ( x ) ~is finite and bounded above by 9
a constant depending only on v, H , q, and I/u II L!(a,. Remark 1: If, for n > 3 , wereplace conditlon (13.2) for the b, by
the assumption
-
(13.14)
2
then inequalities (13.13) will be satisfied for sufficiently small p, more precisely, for values of p that satisfy not only the condition (13.12) but the condition (13.15) This is true because, on the basis of H’dlder’s inequality and the inequality (13.11) with q = n, the inequality
is valid for the t e r m
However, in the present case, the quantity depend also on
and the choice of p
Thus, if we replace the assumption (13.2) regarding the B , ( x ) for n > 3 with the assumption (13.14), then the assertions of Theorem
CONDITIONS UNDER WHICH GENERALIZED S O L U T I O N S SELONG
201
13.1 remain in effect except that the values of essential maxlu(x)I PI
now depend also on A(p). A s the examples in Section 2 of Chapter 1 show, this dependence is a genuine dependence. Remark 2: If we relax the assumptions regarding the inhomogeneous terms f i ( x ) and f ( x ) in the hypotheses of Theorem 13.1, we can, by using (13.4) with y = uCmC2, where cm=2(*)m-
(for m = 0, 1, following:
...),
1,
and also inequality (13.13) with q= n, prove the
IffiEL,(Q)andfEL,(&),where 2
Iff
nr n-r
-};ns
n--2s
E L, ( Q ) and f E L -, ( Q ) for some p > 0 , the integral 2
i s finite. 14. CONDITIONS UNDER WHICH GENERALIZED SOLUTIONS IN W:(Q)BELONG TO C,,= Let us show that, when conditions (13.2) and (13.3) a r e satisfied, an arbitrary generalized solution u ( x ) in W i ( Q )of Eq. (13.1) belongs to the class C , for some a > 0. To do this, we note that, from the proof of Theorem 13.1, more precisely, from inequality (13J3), we get Lemma 14.1. Suppose that conditions (13.2) and (13.3) aye ~
satisfied and that u ( x ) i s a bounded generalized solution W : ( Q ) of Eq. (13.1). Then, u ( x ) belongs to the class B2 (a, M , y, m,l/q), where M = essential max I u ( w ) l and y i s a constant determined by P the quantities v, p , and q in conditions (13.2) and (13.3). From this lemma, Theorem 13.1, and Theorems 6.1 and 7.1 of Chapter 2 regarding functions in the classes d,,we can easily derive Theorem 14.1. Suppose that u ( x ) is a generalized solution in W:(Q)of Eq, (13.1), the coefficients and inhomogeneous terms f i and f of which satisfY conditions (13.2) and (13.3). Then, u ( x )
202
LINEAR
EQUATIONS
belongs to the class C,, (?), where Q' i s an arbitrary interior subregion of the region &, and the norm 1 ~ 1 o~ , .is bounded f r o m above by a constant depending only on the quantities v , p , and q in conditions (13.2) and (13.3), llu/l,,2(Q,,and the distance f r o m 52' to the boundary S of the region 8. The exponent a > 0 i s determined by these same quantities. If we assume that essential max)u (x)J= M < 03, then 11 (x) E Cq, (Q) for some a > 0 determined only by the quantities v, p , q, and hl. I f a portion S , of the boundary S satisfies condition ( A )and i f u IsI E C", (S,),then, for an arbitrary subregion 8 , of the region P that lies at a positive distance d from S \ S , , the function u (XIE C,,, @,), and the norm 1ul0, 9, is bounded from above by a constant depending only on v, p , q, I J U ( I ~ ~ , ~ , ,( u I . ~ , s,, B, d , and the constants 0, and a, in condition (A). These quantitzes also determine a > 0. In particular, i f the entire boundary S satisfies condition ( A ) and if I I ( , E C,,(S), then I I ( X ) ~ C ~ , ~and ( G )the norm ( L I ( ~is, ~bounded f r o m above by a ~ , , the constants constant depending only on V , p, q. ~ ~ U ~ JI U ~I ~ , , ~~ , pami 0, and a , in condition (A). Remark 1: Here, just a s with Theorem 13.1, condition (13.2) regarding the b , ( x ) can be replaced by the condition (13.14), but the values of the norms Iu 1 9, will then depend on A (p) a s well a s the quantities mentioned. Remark 2: The assertions proven in Sections 13 and 14 a r e consequences of the fact that the functions u ( x ) being investigated satisfy inequalities of the form (13.13). In these two sections, we have shown that functions u (x) that satisfy the identity (13.4) satisfy these inequalities also. However, if a function u ( x ) E W: (9)satisfies not (13.4) but, for example, the inequality 0
~
cL,
for all known positive ~ ( x in ) w:(Q),it also satisfies inequalities (13.13) and therefore its maximum can be bounded from above in terms of the quantities indicated in Section 13. In Chapter 5, we shall give another class of gLsubso1utions,99 that is, functions that satisfy not the equations or the identities substituted for them but some system of inequalities and we shall prove their HGlderness. A slight generalization of this class is the following class of ''subsolutions:" Suppose that u (x) E W i ( Q ) , that essential max I u (x)l,< M , and d that u ( x ) satisfies inequalities (14.1) for all q ( x ) that satisfy the conditions - M ,< u (x) + t q ( x ) < M for t E [ 0, 11. Ttturns out that all such u (x, belong to C,,a, where a > 0, and that their norms J u ) , satisfy the same bounding inequality a s do the norms of the
CONDITIONS F O R BOUNDEDNESS
203
generalized solutions of Eqs. (13.1). The proof of this is analogous to the proof of Theorem 4.2 of Chapter 5.
15. CONDITIONS FOR BOUNDEDNESS OF rnaxlVirl AND IuxiI FOR GENERALIZED SOLUTIONS IN Wi (52)
Let u s see what are sufficient conditions regarding the coefficients and inhomogeneous t e r m s in Eq. (13.1) for an arbitrary generalized solution of it in W:cQ!, to belong to Cl,a(Q). It follows from the results stated at the end of Section 11 that there exists an a priom' bound for llulj w;(p,, where q > n, in t e r m s of lluilL2(Q),the constants v and I-( in (13.2), the constant I-( in the inequality
and the constants characterizing the surface S c W i and the norm 11y111w;ce,of the function p ( x ) which determines the boundary values of u (XI. The bound for (lullwg,pxy with q > n, gives in turn a bound for
-
lull. =, *, where a = 1 n/q (cf. Theorem 2.1 of Chapter 2 on the inFrom these a priori bounds jection of W: (Q), where q > n. inC,, and the theorems proven above regarding the solvability of the first boundary-value problem in the spaces C2. and W i , it is easy to show the validity of Theorem 15.1. If the coefficients in Eq. (13.1) satisfy conditions (13.2) and (15.1), i f !c is a region of the class W ; , and i f '9 ( x ) E W i ( Q ) , then an arbitrary generalized solution u ( x ) of Eq, (13.1) in W i ( 2 ) n / q , and the quantities belongs to W i ( Q )nC,,=(G), where a = 1 II u 11 w;(,)and (11 1 aye bounded from above by a constant depending only on n , q. v. p. JjulJL,,p,, llpll w;(B),and the region c?. This theorem is proven by following the same procedure a s in the proof of Theorem 10.1. We can obtain an a priori bound for ~ u ~by, a, different ~ procedure, similar to the one used in Sections 13 and 14 to derive bounds for max IuI and 1u1(,,. Specifically, let u s differentiate Eq. (13.1) with respect toxR,for h = 1, n, and let u s write the result in the form o1
(a)).
-
,,
R,
...,
k=l,
where
.... n.
204
LINEAR EQUATIONS
and 8; denotes the Kronecker delta. Letustreat the set of relations (15.2) and (13.1) a s a system of equations in theu,, where k = 0, 1, n. Its characteristic feature is that the principal part, con, i s diagonal and is the sisting of terms of the form - a,,-dF dxl same in all equations of the system. We have succeeded in investigating such systems with a s great detail a s a single secondorder equation. Chapter 7 is devoted to these systems. There, we derive, in particular, a bound for the maximum of
...,
"(
")
which, when applied to the present case, yields the desired bound for max IVuJ, where Q'cQ.A bound for (Vul close to the boundary 9' does not follow immediately from the bound given in Chapter 7 for max IuI since the value of max lul in Chapter 7 is bounded in terms 9 of the value of max I u J , which we do not know for the vector
...,
I 1
S
du
only the quantity dn is unknown in S advance). To obtain such a bound, we need to examine further certain considerations regarding the boundary spheres. For lack of space, we have been obliged to omit them. However, they can be reestablished when we have read Section 16: The bound explained in that section for (Vul near the boundary between two media is very close to the bound for )Vu(close to S. Bounds for the quantities I uXi 9 , , where Q ' c P also follow from the results of Chapter 7. However, they can be derived from the theorems in Section 14 of the present chapter. ln fact, if the quantities M = niaxlul and M, = maxlVuI are already known, the reP 9 lationship (15.2) can be regarded a s the equation u = ( u , u,,,
u X n ) (actually,
(15.3)
...,
for u k for which all conditions of Section 14 a r e satisfied. Conn, the conclusions of sequently, for & = u x k y where k = 1, Section 14, namely, Lemma 14.1 and Theorem 14.1, a r e also valid. They provide a bound for I uxi where Q'cQ.To get a bound for
205
CONDITIONS F O R B O U N D E D N E S S
I uxr I(n)
close to any portion S , of the boundary S , we straighten that
portion out, introducing new coordinates. Without loss of generality, let u s assume that S, lies in the plane (x,, = 0) and that 8 is adjacent to it from above, that is, onthe side x , > 0. For the derivatives uxr, where T = 1, . . . , n- I, the boundary values on S , are known. Therefore, we can apply to uxTthe second portion of Theorem 14.1, dealing with a bound for lux I where T = I , . . . , n - 1, close to I
(a), QP'
S,. (Here, QP is the half of the sphere K, with center on S , belonging
to
n.)
for
To get a bound for
T = 1,
. . ..
n - 1, in
I ux, I n , , we note that P . . .) implies
?$(Qp.
membership of
u.~-,
n
F r o m this and Eq. (13.1), we get
A s was shown in Lemma 4.1 of Chapter 2 , these inequalities imply that u,, belongs to C,),~ ( 9and ~ they ) enable u s to get the desired bound f o r I u l'n lo,9. Thus, we have a second procedure for deriving bounds
for max lbul and I n ,
I
i (a)
for solutions of Eqs. (13.1). On the basis of
it we have proven Supplement to Theorem 15.1. Suppose that conditions (13.2) and (15.1) a r e satisfied for Eq. (13.1) and suppose that only a portion S, of the boundary S i s a s u r f a c e i n the class W i . Then, an arbitrary generalized solution u ( x ) in W : ( Q ) of Eq. (13.1) will belong to C,, @,), where 9 , is a subregion of Q lying at a positive distance from S \ S, if it coincides on S , with the function 'p ( x ) E W i (9).lts norm ( u (,, P, 11,
is bounded from above by a constant depending only on
quantities n,
v,
p,
the
and q in conditions (13.2) and (15.1), on the
norm llQll W ' ~ ( Q ) ' on the distance from 8, to S \ S,, and on the properties of the surface S,. 16. DIFFRACTION PROBLEMS
Ordinarily, the steady-state diffraction problems that we are considering are special cases of the problem of determininga
206
function
LINEAR EQUATIONS
u (x) in
a region Q that satisfies the equation
some condition on the boundary S of the region 8, and the following conditions on a surface 'l of discontinuity of the first kind for the coefficients ui,: (16.2)
Here, p ( x ) i s any positive function with a possible discontinuity of
I
dU the first kind on ,'I dN = u i j u x .cos (n, xi),where n is the normal to I
,'l and the symbol [u]denotes the jump in the function v a s it crosses .'l The surface 'l is contained in s1. A l l the given functions a r e assumed to be sufficiently smooth outside it. The functions 'p, and 'p2 a r e known functions. The unknown function u undergoes a discontinuity determined by conditions (16.2) on crossing I?. Outside I', it must satisfy Eq. (16.1). Most commonly, we encounter the case in which the surface I' partitions the region Q into several sub62,, where S2 = Q, U 8, . . . u 9,. This corresponds regions Q , , Q2, to physical problems in which there a r e several media of different substances [in each of which the equilibrium state is described by some equation of the form (16.l)l applicable to it and in which u obeys the relations (16.2) upon crossing from one medium into the other. Let us consider this case. Here, a s throughout the book, we consider only the case in which the region 8 is bounded. However, the greater part (including the fine points) of the reasoning that we follow with regard to the investigation of the differentiability properties of generalized solutions of diffraction problems a r e independent of whether the region Q is bounded or not. Therefore, it is immediately applicable to the case of arbitrary unbounded regions. ln order to make certain inconsequential simplifications in our presentation, let us assume that a in (16.2) is equal to 0. Suppose that the function n ( x ) satisfies the first boundary condition on the boundary S of the region &. (The cases of the other boundary conditions can be considered in a similar manner.) This boundary condition and condition (16.2) can be reduced to homogeneous boundary conditions by making the substitutions v (n) = u ( x ) - 'p ( x ) , where 'i. (x) is such that it coincides on S with u (x>l and undergoes jumps on' I that a r e determined by conditions (16.2). Suppose that we have made this reduction, and that u ( x ) satisfies conditions (16.1) in the regions Q,, . . ., 9,and the conditions
...,
DI F F R A C T I O N P R O 6 LE M S
20 7
(16.3)
on the surfaces S and.'I The problem (16.1), (16.3) can be reduced to finding a generalized solution in the space Wi(S2) of the first boundary-value problem for #la single" equation of the type (16.1) though with, generally speaking, coefficients that a r e not differentiable throughout the entire region P, that is, to the problem considered in Sections 4-5. This equation i s the following:
It is obtained by multiplying (16.1) by p and formally carrying out the operations of differentiation and regrouping of the terms. For the moment, these operations must be regarded a s purely formal ones. Let u s suppose that p ( x ) is a bounded function exceeding some positive constant p 0 and possessing generalized first derivatives in each of the regions Q,, for I = 1, t, that are g-summable, where q > n, over 9,. The coefficients u,,, b,, and a and the function f a r e assumed to satisfy the same conditions a s in Section 13, namely, conditions (13.2) and (13.3). We shall refer to a generalized solution in W:(Q) of the first boundary-value problem with homogeneous boundary condition for Eq. (16.4) a s a generalized solution in W:(Q) of the diffraction problem (16.1), (16.3) or, more briefly, a s a generalized solution of the problem (16.1), (16.3). In other words, we shall say that a
...,
function
u ( x ) belonging
to lb: (Q) and satisfying the integral identity
@:
for arbitrary E ( x ) E ( Q ) is a generalized solution in W: (Q) of the problem (16.1), (16.3). Suppose that the function u ( x ) satisfies all conditions of the form (16.1), (16.3). More precisely, suppose that u ( x ) belongs to Wi(52,) for each of the regions Q, (where 1 = 1, t ) , that, in each Q,, the coefficients a,, ( x ) a r e differentiable and belong, for example, to L,(Q,), where q > n, and that 11( x ) satisfies Eq. (16.1) for almost all x in Pi (where 1 = 1, t ) and conditions (16.3) for almost all points in r and S (in the sense of (n 1)-dimensional measure). Then, it is easy to verify that under these Conditions u ( x ) also be-
...,
...,
-
longs to the class Wi (&)andsatisfies the identity (16.5). Specifically,
208
L I N E A R EQUATIONS
...,
since u ( x ) belongs to Wi(Qi), for I = 1, t , and since [ u l l r = 0, it follows that u ( x ) E W: (Q). This, together with the condition u Is = 0 ensures that
u(x)
will belong to W:(9). Now, let us multiply Eq.
(16.1) by pE, where E E Wl (II) and , integrate over 8,. If we then integrate the first term by parts, taking p i d x a s “dv,” we obtain the identity (16.5). All the boundary integrals, which appear explicitly when we carry out the integration by parts, disappear: on I’ be-
and on S because El, 0. The converse is also CEII, true; specifically, if the given conditions of the problem possess the
[
cause p - -
=O
=
differentiability properties just indicated, and if the function on u (x)
belongs to
lb: (Q) and
W i (Qi),for I = 1,
...,t, and satisfies the
identity (16.5) with E E Wi(Q), then u ( x ) satisfies Eq. (16.1) (for almost all x in Q) and the conditions (16.3) (for almost all points on I’ and S). These considerations indicate that it is possible to replace the classical formulation of the diffraction problem in the form (16.1), (16.3) with another generalized formulation in the integral form (16.5). But when the diffraction problems a r e posed in this last way, they a r e reduced to finding a generalized solution of Eq. (16.4) in the class @:(Q). In the preceding sections, we investigated such a problem and proved its Fredholm solvability. We now formulate a portion of this result a s Theorem 16.1. Suppose that the following two conditions are satisfied: n
n
~ Z E : , < a r j E i E j , < P ~ E V~>,o , i=l t=l
I=1,
o
(16.6)
... t , q > n .
Then, the existence of a generalized solution w:(Q)of the diffraction problem (16.1))(16.3) is implied by the uniqueness of the problem. If I l f l l Lq (O)<
03,
2
(16.8)
then every generalized solution in w:@) of the problem (16.1), (16.3) belongs to C,,,(8)f o r some a > 0 . If, in addition, the boundary S satisfies condition (A), then u E C“,
(a).
20 9
DIFFRACTION PRO6 L E M S
This theorem is a direct consequence of the theorems in Sections
4, 5, and 14. In those sections, we gave sufficient conditions for the Dirichlet problem to have no more than one solution in W:(Q). Let
us find the conditions under which the generalized solution u ( x ) of the problem (16.1), (16.3) is a classical solution, more precisely, under which it will belong t o Cz, (Q,) n C,, (a,)for each 8,and hence will satisfy all the conditions of the problem in the form (16.1) and (16.3) directly. As always when we a r e studying elliptic equations, the investigation of the differentiability properties of their solutions can be carried out locally, over small regions. It follows from Theorem 12.1 that, if the functions al,, dai,/dxk, bi, a, f, p , anddp/dxi belong to the classes Co, ( Q l ) for 1 = 1, . . ., t , then the solution u ( x ) belongs to the class Cz,B ( Q l ) for 1 = 1, . . ., t. Furthermore, if the function u ( x ) vanishes (or coincides with a function in C2, on some portion S' of the boundary Q , that is contained in S but has no points in common with I' and if s' Cz, then u ( x ) E C2, (Q, u S'). One thing that the theorems proven above fail to give us is information regarding the behavior of the derivatives of u ( x ) in a neighborhood of .'I Let us examine this question for the first derivatives of u ( x ) , that is, for those derivatives of u ( x ) that appear in condition (16.3). In order to make certain insignificant simplifications in the presentation, let us assume that all the coefficients in the operator L and the derivatives dal,/dxk and dp/dxk a r e bounded, that is, that @)
e
@,
(16.9) Let u s also assume that conditions (16.6) a r e satisfied. Let u s take some small portion r,cl' that belongs to Cl, and straighten it out introducing new nondegenerate coordinates y = y ( x ) possessing bounded first and second derivatives with respect to x. In doing this, we do not destroy any of the properties of Eq. (16.1) that we need, and the equation keeps the same form. Therefore, we may assume without loss of generality that, even in the x coordinates, Pl is a portion of the plane x,= 0. Suppose tht the center of the sphere K , is on PI, that i t s left half (forx, < 0) belongs t o Q , , and that its right half belongs to Q,. Let us denote by C ( x ) any twice continuously differentiable function with values between 0 and 1 that is of compact support on K,. Let us show that u ( x )C ( x ) E W i ( K , f l Q l ) , for 1 = 1 , 2. The proof is essentially the same a s the proof of the analogous assertion for generalized solutions in the case in which the leading coefficients have no surface of discontinuity l'. The corresponding a priori bound constitutes the basis of the proof. In obtaining it, we need t o remember that, for the
210
LINEAR EQUATIONS
coefficients q j , there a r e no generalized derivatives of the form throughout the entire region & (they exist only in 52, and 8, separately) but that we still have the remaining derivatives d / d x f , for 1 = 1, n 1. Therefore, we need to obtain an apriom' bound such that the constant in it will be independent of d a f l / d x n . If we look at the derivation of inequality (8.6) in Chapter 3, it is easy to see that this derivation involved the existence of d a f j / d x , , i n & and the fact that these derivatives belong t o L, (Q), f o r q > n. To avoid this, let us consider not
d/dxn
...,
-
but
o r , what is essentially the same, let us consider identity (16.5) with n-1
where C ( x ) is a twice continuously differentiable function with values between 0 and 1 that is of compact support on Up. Let us integrate the first t e r m in this equation over each of the Q,,for 1 = 1, 2, once by parts, transferring the derivative d/L1xk with (uXkC*),, to the remaining factors. Since x k , for k n - 1, is a tan-
gential coordinate with respect to r,, we do not get a boundary integral over r,. There will also be no integral over the boundary of the sphere K, since C vanishes on it. The principal t e r m will be
It is not less than n-1
n
A s we can easily verify, none of the remaining t e r m s exceeds in
DIFFRACTION P R O B L E M S
21 1
absolute value the quantity
where E is an arbitrary positive number and c. ( I V C J ) is a known constant depending on max 1 VCI. Therefore, from our equation we get the inequality
KP
(16.10)
We derived this inequality under the assumption that the derivatives uxkxl are square-summable over K,. We could have derived the inequality without making this assumption by considering not the second derivatives of u but the difference quotients A/Axk of the first derivatives uxf and then finding a bound for the integral
To do this, we need to take E(x) in (16.5) in the form I.
.
)z1
k=l
(-&
c2).
When we have found a bound f b (uniform over the sphere A x , ) , be can then conclude that the derivatives uxkxi, for k ,< n - 1 and 1 4 n are square-summable over Up,,where p’ < p. It is possible to prove that the integral in the left-hand member of inequality (16.10) is finite in another way. This proof r e s t s on the existence of an a priori bound of the form (16.10) for sufficiently smooth functions u ( x ) and it uses the fact that the constant c ( I VCI) in (16.10) is independent of the magnitude of the derivatives dal,/dx,. The general procedure is the same as the procedure used in proving Theorem 10.1: the coefficients in Eq. (16.1) and the inhomogeneous t e r m f (x) a r e approximated by functions sufficiently smooth that all solutions of the approximating equations Lmu=fm,
m=I.
2,
....
(16.11)
21 2
LINEAR EQUATIO N S
are functions belonging to C2,*. Out of all the solutions of Eqs. (16.11), let us take those that coincide on the boundary of K, with the solution u ( x ) of the diffraction problem (16.1), (16.3) that we are studying. For spheres K, of sufficiently small radius, this requirement defines for each Eq. (16.11) a unique solution urn( x ) . We c a r r y out successive approximations such that the sequence p}converges to f in L 2 ( K p ) ,such that the sequences of the coefficients u?,, I!$, and urn,which are bounded in absolute value by some constant, converge almost everywhere to u l j ,b,, and a , respectively, such that the lowest characteristic value of the form a;€,€, will be
1 =I,
da;
no less than vo > 0, and such that the quantities f o r k ,< n - 1 , will remain uniformly bounded. For uYi, by. a'" and f"',we can, for example, take the averages of u,,, b,, a, and f, respectively, withaveraging radius equal to l / m . One can prove, just as in Section 10, that the functions u m ( x ) are uniformly bounded in the norm W:(K,) and that the sequence of these functions converges to u (x). Furthermore, as inequality (16.10) shows, the quantities /,, c(urn)will also be uniformly bounded. Therefore, the quantity I,, ( u ) is finite. Thus, we have shown that the derivatives uxkxi, for k < n - 1 and 1 ,< n are square-summable close to rl. It follows from Eq. (16.1) and inequality (16.10) that the quantity u; is summable in n n
each of the regions K, n 51, and K, n Q,, more precisely, that (16.12) where c ( I VC I ) is, in general, different from the c ( I VC I ) in (16.10) but is still a known constant. Note that, although we write the norm uC in W : ( K p ) in the expression on the left, the function u (and hence the function uC) does not belong to the space W;(K,) since u does not have a generalized derivative uxnxn everywhere in K,. Such a derivative exists only in each of the subregions K , n M, and K , n 9,. Here, the symbol denotes the quantity Inequality (16.12), together with the identity (16.5) ensures that the solution u ( x ) will satisfy Eq. (16.1) for almost all x in K, n Q k ) in each of the regions K , n Q k , for k = 1, 2 and that it will satisfy conditions (16.3) on for almost all x on I', in the sense of ( n 1)dimensional measure.
-
213
DIFFRACTION P R O B L E M S
Let us show now that if, in addition to conditions (16.6) and (16.9), we assume that max 1 f 1 is bounded, then the product uC will 9
belong t o C,,,(K,n Qk), for k = 1, 2. We first show that 1 V u l i s bounded, assuming that we know M = essential max j u 1. KP
For the function E in (16.5), we take the function d?,v/d.w,, for s < n - 1, where ~ ( x is) at the moment a sufficiently smooth function that is of compact support in K,. When we substitute it into (16.5) and integrate the first t e r m by parts, we get
Let u s take for K , a sphere that is concentric with the sphere K , chosen above [for which inequality (16.12) has been proven] but with a sufficiently small radius that the derivatives uxix, are squaresummable in the new sphere K,. Then, wemay take for qs in (16.13) an arbitrary function in &;(K,). In (16.13), let us set ys=uxsbrC2, where s , < r t - 1 and C(x)is a smooth function with values between 0 and 1 that is of compact support on K,, and
where
6 is a positive number not exceeding unity (which we shall choose
sufficiently small in what follows), and Nis a large positive number, which we shall let approach a. On the basis of the conditions (16.13), we have [uxs]I = 0 for s ,< n.- 1 and r,
This, together with the fact that u ( x ) belongs to W i ( K ,n 8,) for k = 0 1
1, 2, ensures that -qsbelongs to W 2(K,).Let us substitute this function
214
LINEAR EQUATIONS
q,, into (16.13) and sum the resulting equations with respect to s from s = 1 to s = t z 1. The principal terms generate the following two terms:
-
Let us represent the expression
in the form
1 2
x n
paijrbr-'bxf6x,C2-
In the equation that we a r e considering, let us transpose to the right side all terms except
Let us find a bound for this expressionfrom below and let us find a bound for the terms on the rightfromabove. In this way, we arrive at the inequality
n-1
n
(16.14)
DIFFRACTION P R O B L E M S
21 5
Let us use Eq. (16.1) to express the derivative u ~ , in . ~t e~r m ~ s of the derivatives uxsxf, where s ,< n - 1 and i ,
in the t e r m s containing I Vb I since, as follows from the definition of b ( x ) (we recall that 6 ,< l),the inequality
is valid, where v and 1-1 a r e the v and I-( in (16.6). By virtue of all this, from (16.14) and (16.1), we derive the fol1owi:ig chain of inequalities:
n-1
n
216
L I N E A R EQUATIONS
Here, E is an arbitrary positive number. For future use, we take only r , < r o = [ n / 2 ] numbers 6 and E sufficiently small so that
-1.
We choose the
Then, after collecting similar t e r m s in (16.15), we have
Using Eq. (16.1) for the derivatives uxnxn,we see that this representation of uXnxnand inequalities (16.17) yield the inequalities that we need:
21 7
DIFFRACTION P R O B L E M S
The constants c and cI that appear in inequalities (16.14)-(16.18) a r e constants determined by quantities known to u s and a r e independent of the number N that appears in the definition of the function b ( x ) . In addition to (16.18), we need yet another inequality, one that will be valid for an arbitrary function u ( x ) E W : (K,n Q*), for k = 1, 2, such that essential max 1 u I = M that satisfies on
rl conditions
<
03
KP
(16.3):
1
(16.19)
To prove inequality (16.19), consider the integral
where n- 1
n
and, a s before, b ( x ) is equal to
Let u s regard J as the sum of integrals over the regions K, r l Q, and I<, n S!, in each of which we integrate by parts, transferring the
218
derivative
LINEAR EQUATIONS
d/dxi
with uX1to the remaining factors. This yields
The surface integrals over ,'I that result from this cancel each other out. The integral over the boundary Upvanishesby virtue of the factor 1. Let u s use Cauchy's inequality in the form (1.2) of Chapter 2, the boundedness of
and the equivalence of the quantities
that is, the validity of the inequalities
to find a bound for the right-hand member of Eq. (16.20). In using Cauchy's inequality (12), Chapter 2 , to bound the right-hand member of Eq. (16.20), we need to assign small E to t e r m s of the form I V u I'vb'Cz and t e r m s equivalent to them. h this way, we obtain from (16.20)
21 9
D I F F R A C T I O N PROBLEMS
+El
Vb12i2+
vu126'+'C2+;b~-lI
+ 1 v u pvb'c2+ E
1
; vb'
I
I vc (2
dx.
If we choose E sufficiently small and collect similar t e r m s , we obtain (16.19). Inequalities (16.18) and (16.19) make it possible to determine that the integrals in the left-hand members of (16.18) and (16.19) a r e uniformly bounded both with respect to the number N in the definition of b ( x ) and with respect to u ( x ) if we know that II u II w;(Kp) ,= PI < 03. To see this, suppose that
..,
Let us construct a sequence of spheres UP,, where P , ~= p/2 + p/2"', for rn = 1, 2, that a r e concentric with Up. For r = 0, inequality (16.19) yields a bound for the integral
and hence for
in t e r m s of
that is, in t e r m s of p1 and
p.
2 20
LINEAR EQUATIONS
For r = 1 , inequaltiy (16.18) gives a bound for the integral
in t e r m s of
but, for arbitrarily large N , this last expression does not exceed
which is a quantity that is known to us. Thus, in particular, the integral
is finite and bounded from above by a known constant. By considering in succession the inequalities (16.18) and (16.19) for r = 1, 2, ro and letting N approachco, we can prove that the integrals
...,
a r e bounded. The constant c ( r o , l / p ) depends on r(, and l / p , the constants in (16.6) and (16.9), max I f 1 and IIUII~;(~,). K.
Let us now prove the boundLdness and Holderness of the tangential derivatives ux, for s ,< n - 1. In the identity (16.13), we set qs (x) = C2 (x) . max ( u X s(x) - k . 01 where k is an arbitrary number and :(x)is a smooth functionwithvaluesbetween0 and 1 that is of compact support on K,. We now take the sphere K,arbitrary though situated within that sphere for which the boundedness of the integrals (1.21) is established. Let us denote by A k , the set of points x in K,
22 1
DIFFRACTION PROBLEMS
at which U , , ~ ( X ) > k. Of course, this set depends on s but, for brevity, we omit the subscript s. If we substitute the function qs into (16.13) and make the elementary estimates, a s we have done several times before, we obtain
+ c ( J (1 + I vu
12)”
dx)’
rnes
,-I p
dk,p.
Ak,p
If we choose p so that 1/p <2/n, that is, so that 2p > nand remember that inequality (16.21) implies 1-summability of I V u I, where 1 = 2r, = 4 = 2 [ n/2] = 2 > n, it follows from inequality (16.22) that A
/I
(uxs - k)* I VC 1’ d x -t
Vuxs (‘C2 d x .
k, P
(16.23)
k. P
Analogous inequalities hold for the s e t s Bk,?, where ux, < k . On the basis of Lemma 5.4 and Theorem 6.1of Chapter 2, we conclude from these inequalities that max Iux,,I is bounded and uXs satisfies a Halder condition in an arbitrary sphere Up,, situated in the interior of the sphere Kp, - for which inequality (16.21) has been estab2
lished. It remains to show that lux,( i s bounded and that u,, satisfies a Halder condition in K,. n &, and K,. n 51,. Let us take an arbitrary sphere K,, in which the quantities max I uxs[ and 1 uxs. 1 Kp:
for s < n
- 1 , are known.
K,
If we set k = min uxs in (16.23) and assume KP
that the spheres K, to which (16.23) applies belong toK,,, we conclude from (16.23) on the basis of the Halderness of uxs that
II
vuxs12 C2dX 4 c2 [P
n-2+2a +
PR
(1
1
--p)Is
c3pn-~+~
KP
where p=min(2a.
2--
“1 > o .
P
From these inequalities and Eq. (16.1), it follows that n
(16.24)
222
L I N E A R EQUATIONS
From inequalities (16.24), which a r e valid for arbitrary spheres K, contained in Kp, of arbitrarily small radius p (as Lemma 4.1 of Chapter 2 asserts), it follows that Iuxn!i s bounded and ux, satisfies a H6lder condition in a sphere Kp-that is concentric with K,,, where p” < p’. In this way, we study the behavior of the derivatives uxi close to the surface of discontinuityl’,. Inessentially the same way, we can study the behavior of uXi in a neighborhood of the intersection n S. Let us formulate the result that we have obtained a s Theorem 16.2. Swpose that 8 i s the union of Q,, . . ., 8,and the boundaries I?,, separating them, where I, j , = 1, t. Suppose that the boundary S of the entire region 8 satisfies condition ( A )and that the coefficients in L. p ( x ) , and f (x) satisfy the conditions
...,
and
Then, a generalized solution u ( x ) of the problem (16.1), (16.3) in the class W;(Q) belongs to the classes Co.m(G)and C?,l ( Q R i , where k = 1, t. I f the ri, are surfaces in the class C,, ,, then u ( x ) has first derivatives that are continuous in the sense of Hzlder up to with the possible exception of points of junction of two OY more of the surfaces I?ii and S. It satisfies all conditions of the problem in the form (16.1), (16.3) and i s a classical solution of the problem (16.1), (16.3). If the boundary S belongs to the class C,,,(resp. the class C2.p), then ii ( x ) has continuous first (resp, second) derivatives in the sense of HElder up to S except possibly at points of junction of S and P l i .
...,
This theorem answers the question a s to the existence of a solution of the problem (16.1), (16.3) in its classical formulation. The generalized formulation of the diffraction problems for equations of various types (expounded in the present section for the elliptic case), the possibility of reducing these problems to the problem of finding generalized solutions “with finite energy integral” (in the present case, generalized solutions in the space W: (Q)) for a single equation with discontinuous coefficients, the consequent applicability of methods and results regarding the existence, evaluation, and investigation of these generalized solutions with finite energy integral may be found in [73] by one of the authors of the present book. Reference [73] also contains a proof of the applicability of the method of finite differences to the actual finding of solutions of
THE CASE OF T W O INDEPENDENT VARIABLES
223
these problems. Furthermore, in that article and in others by the same author (cf., for example, [74, 75]), it is shown how one can investigate the differentiability properties of generalized solutions of general elliptic equations and, in particular, diffraction problems in the spaces W:(&), where 1 2. (These investigations a r e carried out in detail in 176, 771.) However, it was noted there that the results obtained by this method regarding the classical solvability of the problem (16.1), (16.3) for arbitrary n are quite crude in the elliptic (and parabolic) case: F o r us to be sure on this basis that the problem (16.1), (16.3) has a classical solution, we need to require that the coefficients in the operator L and the surfaces,,'l have bounded derivatives of order at least [n/2] + 3. The methods that we have expounded for investigating generalized solutions of the problem (16.1), (16.3) in the spaces Cl,@ a r e considerably finer. We mentioned their applicability to diffraction problems for elliptic and parabolic equations when we first constructed these methods in the reports [5], [15], etc. We note that, up to the time of publication of the article [ 731, a thorough study of diffraction problems had been made only for equations with constant coefficients (we are referring to the case n > 2), and the solutions of these equations had been sought in the form of potentials or infinite series. In this equation, so-called weighted integral equations were obtained to determine the densities of the potentials. (In the more fortunate cases, these integral equations were reduced to Fredholm equations of the second kind.) Infinite algebraic systems were obtained to determine the coefficients in the series. A proposition similar to Theorem 16.2 was also proven in a recent article [go]. The method of proof is different from the one given here.
>
17. THE CASE OF TWO INDEPENDENT VARIABLES The results expounded in the preceding sections a r e valid for an arbitrary number of independent variables, including the case ri = 2. However, certain things a r e true of the case n = 2 that a r e not true for ti > 2. Some of these were pointed out in Section 2 of Chapter 1. Among these a r e the following: Suppose that u (x) is a solution, in a region 51, of the equation (17.1)
the coefficients in which satisfy the conditions
2 24
LINEAR EQUATIONS
If the solution u vanishes on the boundary of the region Q, then the norm 11 u )I w;(p) for it is bounded from above by a constant depending only on the constants Y and p in (17.2) and (17.3) and the norms I l f / l L , ( Q ) and I I U I I ~ , ( ~ )This ~ fact was established by Bernstein [22] for solutions of Eqs. (17.1) with ai = a = 0 in a circle. Combining the device of preliminary transformation of Eq. (17.1) given by Bernstein with the device of transformation of the contour integral resulting from integration by parts and the bound for it given by one of the authors [55, 101, we shall now prove the assertion made for an arbitrary region Q. Furthermore, we shall weaken as much a s possible the assumptions regarding a, and a. Specifically, we replace condition (17.3) by the assumptions ;iailILq(Q).
llaltL2(Q~
1(1+lPIZ).
..
(2.1) (2.2) (2.3)
Here, b ( x . u. p ) = ( b ' ( x , u, p). ., b N ( x . u. p)), and E ( M ) is a sufficiently small quantity determined only by n, N , M , v ( M ) , and p(M)
410
QUASILINEAR S Y S T E M S
in (2.1) and (2.2), and P ( p , M)+ 0 a s IpI -+a. From what was said above, it is clear that the restrictions (2.1)-(2.4) a r e brought about by the nature of the problem. Suppose that u(x) is a solution of the system (0.3) that belongs to the class C,(51). Let us suppose that we know max I u I = M and the Q
constants v ( M ) , p ( M ) , E ( M ) ,and P(p, M ) in (2.1)-(2.4). Let us show that it is possible to give a bound for 1 u In. in terms of these constants for some a > 0 and arbitrary Q'cQ.To do this, we need only show that u belongs to the class Q 8
the parameters of which a r e determined by quantities known to us. Without lose of generality, let us assume that 0 < u1 < 1, where 1 = 1, N. We introduce the 2iV functions (9:(u)= lONu'+v, cp'_ (u)= lON(1 - d ) + v ,
...,
where N
V=
2 (u')*,
I-=
1
1 = 1,
. . . , N.
...,
Let us show that the functions w ( x )=:Q (u(x)), for 1 = 1, N, satisfy the inequalities nf the definition of the class B?. Let us take the scalar product of the system (0.3) and the
vector -q(x)E i i ( Q and ) then integrate over 8. Tfwe then integrate by parts on the left, we obtain
+
10iVe') @ ( x ) , where @ (x) E $; (Q ) with Q'cQ Let us take q = (2u and e' is a unit vector in the space of N-dimensional vectors with lth component nonzero. Let us substitute this vector q into (2.5), and let u s write the result in the form
where
A BOUND FOR
I U In,
41 1
B
In (2.6), let us set Q, ( x ) = { 2 ( x ) max (w'+ ( x ) - k ,
0)
where C ( x ) is a smooth function of compact support taking values between 0 and 1 on the sphere KO. Then,
Here, A k , pis the set of points x in the sphere K p at which w L ( x ) > k . Let u s find a bound for the left side from below, using the condition of ellipticity (2.1) and let u s boundthe right side from above, using the assumptions (2.1)-(2.4). We use the smallness of the quantity € ( M ) + P ( p , M) in (2.3) for large values of IpI in order to bound the right side of the inequality
1 c y I ,<
+ 1 ON)
I E (M)+
p (P. MI1(1
+I P 13
for all values of p in t e r m s of v I p I*+ c , where c1 i s some constant. Clearly, for this it will be sufficient if (211.I
+ ION)
E
(M)
<
V.
(2.8)
Here, the constant c1 will depend onP(p, M). A s a result of all these boundingprocesses, which are analogous to those that we carried out more than once above (see, for example, Section 1, Chapter 6), we a r r i v e at the inequalities
..., N ,
1=1,
for all k satisfying the condition max (w: KP
- k ) < 6.
where 6 is a sufficiently small positive number (though fixed in t e r m s of quantitites known to us). Inequalities corresponding t o (2.9) are also derived for w ! , where 1 = 1, N . From inequalities, we get inequalities (8.3) of Chapter 2, which appear in the definition of the classes %f'.Thus,
...,
412
QUASILINEAR SYSTEMS
we have shown that u belongs to the class B:N(Q, M i ,
. . ., 6,
0)
and its parameters a r e determined by quantities known to us. To get a bound for 1 u I,. throughout the entire region P, we need to show that
Let us take an arbitrary sphere K , that intersects the boundary S. In this sphere, (2.7) is valid for w = w:, and hence so is (2.9) provided k satisfies the conditions k
> max w ( x ) , k > inax w ( x ) -6. K,ns
Kpne
This means that u E BiN @, . . . , 0). From this and the properties of functions in the classes %;N(a,
. . .), BiN(jZ, . . .)
we get
Theorem 2.1. Suppose that u ( x ) 6C, ( Q ) and thatu(x) satisfies the system (0.3), that inequalities (2.1) -(2,4) are satisfied by the coefficients in that system for x E 32, 1 u I M = niax 1 u ( x ) I and arbitrary 9 p, and that the constant E ( M )in (2.3) satisfies inequality (2.8). Then, for an arbitrary interior subregion Q'cQ, there exists a bound f o r IuI.,,, dependirg only on n , N , M , v(M), p ( M ) , ~ ( M ) , a n dP ( p . M)in (2.1) - (2.41, and the distance from Q1 to S. The exponent a i s determined by these same quantities except for the distance from Q' to S. I f , in addition, u beipngs to C,, (a) and S satisfies condition ( A ) , then, there exists a bound for I u. .1 ,in terms of n , N , M , v (M), p (M), E ( M ) , P ( p , M ) in (2.1) - (2,4), the norm I u I s, and the constants a, and 0, in condition ( A ) , The index a is determined by n , N, M , v ( M ) , ~(4 E ( M, I , P ( P . M), p, a,, and 0,.
We note that, just a s in the case of a single equation (cf. Chapter
4), the assertions of the theorem remain valid for generalized solutions of the system (0.3) in the space W:(Q).However, throughout
the chapter, we shall confine ourselves to a consideration of classical solutions only. The theory of generalized solutions of the system (0.3) is constructed in a manneranalogousto what was done in Chapter 4 for a single equation with principal part in divergence form.
41 3
T H E ENERGY INEQUALITY
3. THE ENERGY INEQUALITY AND A BOUND FOR maxlVu) ON THE BOUNDARY Suppose that u q ( x ) in
$i(Q)
Lki(S2) satisfies
the identity (2.5) with bounded
and essential max 1 u I = M. Suppose that conditions
(2.1)-(2.4) hold for the system (0.3) just as in Section 2. Then, for an arbitrary sphere K , , (3.1) where c is determined only by known constants in (2.1)-(2.4), and by n, N , and p. Specifically, in (2.5), let u s take 7=2u (d*-1) L2,
where u=1uI2, where A > 0 is a large numerical parameter, and where C ( x ) is a smooth function of compact support with values in [ 0 , 11 on K,. This yields
If we now remember our assumptions (2.1)-(2.4), including the assumption that E ( M )in (2.3) is sufficiently small and if we choose A sufficiently great, we obtain inequality (3.1) from this equation. Thus, we have proven l e m m a 3.1. Suppose that u E ~
with
qE
k ( Q ) satisfies
the identity (2.5)
+; (Q), essential max I 7 I < oo u essential max I u I = M. P
R
Suppose that the same assumptions hold f o r the system (0.3) as in Theorem 2.1. Then, f o r the function u and an arbitrary sphere Up, inequality (3.1) i s valid with a constant c depending only on n , N , M, p , and the constants in (2.1) - (2.4). Let u s now see about a bound for1 Vulon S. Let u s prove
414
QUASILINEAR S Y S T E M S
l e m m a 3.2. Suppose that u i s a solution of the system (0.3) in C,(8)n C, (Q) that vanishes on S. Swpose that inequalities (2.1) - (2.3) are valid f o r (0.3). Then, max I Vu I i s bounded by a constant deS
pending only on n, N , M = mQ$x IuI, the constants v ( M ) , p ( M ) , E(M), P ( p , M ) in (2.1) - (2.3), and on the boundary S , which is assumed to belong to 0,. By hypothesis, u Is = 0. Consequently,
where d / d n is the derivative along the outer normal to S. Let u s take a point xo on S and an integer r between 1 and N inclusively such that
Suppose, for definiteness, that
Then, let u s take the function w'=u'+ du' (hcase dn
V 3 U '
+[ u
12.
Ixo > 0, we would needtoconsider the function -u'
f~.)
For this function,
It follows from (0.3) that L' (u)
+2
N I= 1
U'L'
(u) = 0,
where L'(u) is the left-hand member of the rth equation in the system (0.3). This equation can be written in the form a,, ( x . u) w;
( 1
- 2a,,ux,uxl
where cr=2b(x.
U.
+b'w' +
u,)u+~'(x.
XI
U,
C'
= 0,
u~).
(3.3)
415
T H E ENERGY I N E Q U A L I T Y
Instead of w r , let u s take the function v' defined by the equation w' = 'p (d).We shall choose the function 'p (y) later, in such a way that 'p'(y) > 0. In (3.3), let u s substituteg'v' in place of wr and let us XI XI substitute y'v; +'p"v;,u;, in place of w ; ~ If we divide the re~
CJ
sulting equation by atj(x.
'p',
i /
we get
U ) V ; ~ ~ ~ +~ 'f"a
.
l j v ; , v ;2l aljuxIuxj+b.vr -~ +7cr=0. 1 L X i 'f
From this, on the basis of the assumptions (2.2) and (2.3), we get
If
E
is such that (2M+ l ) e < v Y .
(3.5)
we can easily derive from (3.4) the inequality
where the constant c is known. Let u s choose the function ' p ( y ) so that
'"( M )-
- _'P'
v
ccq'
>/ 0 ,
where v ( M ) is the v ( M ) of (2.1) and ditions are satisfied by the function v
(MI
V ( Y )=7 In (1
'p (0)= 0.
c
is the
+
c
of (3.6). These con-
Y).
For suchy(y) inequality (3.6) yields - a,jv;,.r
1 1
4 c.
(3.7)
The function v r , like the function ur, vanishes on S and the function
416
QUASILINEAR S Y S T E M S
attains i t s minimum at the same point x,) on S a s does the function d u r / d n . Let us construct a ba r r i e r function + ( x ) , that is, a function satisfying the relations
- a,, ( x . u) qJXiX, < -c.
","" 'p (4= 8 (xo)*
For such a function, we may, for example, take the function
+ ( x )=
me-*@tX'
with .sufficiently large 1 and m and with a twice continuously differentiable function Q, ( x ) possessing the following properties: (1) 4 ( x ) > 0 in 9; (2) I V @ [ > cons t > OinG; (3) the surface Q, ( x ) = 0 contains the point xo. If the region Q lies entirely on one side of the plane tangent to S at the point xo (suppose that the equation of this plane x n = x : ) and if S2 is contained in the half-space (x,>xO,), then, for ( x ) , we may take Q, ( x ) = x n
- x;.
In the opposite case, we would need to transform the region P at the very beginning in such a way that its position relative to the point xo will be as indicated. For the function d ( x ) + qJ ( x ) , we have
- ai/(v'
+qJ)x,xj
< 0.
Consequently, it attains its maximum on S. But,
and, therefore,
On the basis of the relations
and 1=1,
....N
S
A BOUND FOR
41 7
maXl V U ] 8
this gives us the inequality
This completes the proof of Lemma 3.2. 4. A BOUND FOR m a x l V u )
Suppose that u(x) is a solution of the system (0.3). In Sections 1-3, we obtained apriori estimates for I U ~ ~ , ~I l, ~ l I ~ ; ( ~ ) ,maxlvul and s
in terms of max It11 and known quantities. The general outline of P the method of obtaining them is the same a s for a single, secondorder equation with principal part in divergence form, but the analytical facts lying at the basis of the bound for I u I n , (the properties of the classes %?') a r e more complicated. Other bounds for expressions involving u, namely, rnax 1 Vu I and max I Vu I can be obQ'
9
tained in almost the same way a s was done in the case of a single equation in Sections 3 and 4 of Chapter 4. Therefore, we shall point out the salient steps without going into detailed evaluation of all the constants. Thus, we shall prove Theorem 4.1. &@pose that all the conditions of thefirst part of Theorem 2.1 are satisfied. Then, f o r arbitrary 53' c 53 , the quantity rnax lVul i s bounded in terms of the same quantities as lulo,p' in P' Theorem 2.1 andmes 0. If, inaddition, u E C, S E 0,) and u Is 0, then max I V u I is bounded by a constant depending only on n , N , M =
(a),
5
e
mQax IuI , the constants in conditions (2.1) - (2.4), mes 53, and the boundary S . In (2.5), let us set q = -(~,,E)~,,_where E(x) is a function that is twice continuously differentiable in 0 and that vanishes, along with its first derivatives, on S. Let us sum the equations thus obtained with respect to k from 1 to n. After some obvious integrations by parts, we arrive at the identity
where V = 1 Vu. ,1 In (4.1), let us set E = 2V92, where i ( x ) is a smooth function of compact support with values in [0, 11 on the sphere K p c 0 and v" is
418
QUASILINEAR SYSTEMS
....
the sth power of V , where s = 0, 1, The first two terms in the integrand yield positive terms. We leave them on the left side of the equation and find a bound for them from below; we transpose the other terms to the right side and find a bound for them from above, using the assumptions (2.1)-(2.4). This leads to the inequality
(4.2)
where the constant c (s) depends on known constants and the number s (note that c (s)--+ 00) a s s +m). On the other hand, n
KP
k=l
KP
If, for u,,, we take the value of u at the center of the sphere Up, then, on the basis of Theorem 2.3., we have max 1u - uoI ,< cpn.
a
> 0.
(4.4)
KP
If we take p sufficiently small that cc(s)p" < 1, we obtain from (4.3) K PJV"2r2dx.&pa
J [ v ' ~k~- v1 u , , " ' ~ v L ~ 2
KP
1
dx.
(4.5)
This inequality, together with (4.2)yields, for all small p , (4.6)
A BOUND FOR IiIaxIvlll 9
... .
419
They, toInequalities (4.5) and (4.6) are valid for s = 0, 1, gether with the original inequality (3.1), enable us to obtain inequalities
1
VS+'d x
,< c (s,
(4.7)
Q')
P'
... .
for successive values of s , where 8' is an arbitrary subregion of P on s = 0, 1, Our subsequent reasoning is also close to the corresponding reasoning in Section 3 of Chapter 4. Specifically, we take the function
v
w ( x )= ( x ) 22 = I v u 12 c2.
where C(x) is a function a s above on an arbitrary sphere K p c Q and we consider the equation
j
Lu
a
[(w
-h) C~UX,] dx = 0,
A1
where Lu is the left-hand member of the system (0.3) and A , is the set of points x c Q ' at which w ( x ) > h >, 0. After transformations and operations analogous to those of Section 3, Chapter 4, we obtain, by consideration of inequalities (4.7),
with arbitrarily small E (but such that c,+m as E + 0). To obtain a bound for max w , it will be sufficient for E in (4.8) to be less than 2/n. This will be the case if we use (4.7) with s [3n/2] + 1. For such E , it follows from Lemma 5.2 of Chapter 2 that w is bounded from above in P' by a constant depending only on E, c , , and IIWJI~,(~,). From all that has been said, it follows that the first part of Theorem 4.1 is valid. It remains for u s t o find an analogous bound in a neighborhood of the boundary. We already know max I Vu I = M 2 from Lemma 3.2.
A bound for
S
I I U I J ~ ; ( ~ ) is given for the entire
region 51. We need to prove (4.7) for spheres K, that intersect the boundary S, more precisely, for the intersections of K , and 51. For numbers s > 1, we obtain inequalities (4.2)-(4.6) in the same way as we did above if, instead of setting E = 2VY2 in (4.1), we substitute
4 20
QUASILINEAR SYSTEMS
This is permissible since E vanishes on the entire boundary of the region of integration K , n 8. In the case in which s = 0, we define E(x) in (4.1) a s follows:
I
E(x)=
V ( x )4 Mi.
0,
2(V(x)-Mi)C2(x),
M:~V(X),<M~+~.
2c2,
V(x)>/M;+l.
After some transformations and bounding operations analogous to those in Section 4 of Chapter 4 for the case of a single equation, we arrive at inequality (4.7) for 8' = K , n 8,and hence we get the inequalities JV"+'dx,
s=o.
1,
...I
I
rnax Vu I 4 const. €4
This completes the proof of the theorem. Remark: If we assume that we know a bound for I u 17, in Theorem 4.1, then we may assume that E ( M )in inequality (2.3) is an arbitrary number since max I Vu I can be bounded for an arbitrary system of the form (0.1) with quadratic growth of the functions a f ( x , u; p) with respect to IpI. 5. EXISTENCE THEOREMS The a priori bounds that we obtained in Sections 1-4 for solutions of the systems (0.3) L(u)=u,,(x,
U)UXX
Ll
+bi(x.
U,
u J u . ~ , + ~ ( x U, . u,)=O
enable us to investigate the solvability in the large of the first boundary-value problem for (0.3) (and other boundary-value problems considered in Chapter 10). Here, we shall not state the results in the same generality a s in Section 8 of Chapter 4 (they a r e the same a s for a single equation) but shall confine ourselves to one of the specific cases where the parameter T occurs in the system (0.3) and to the case in which it is possible to give an actual bound for max Iul. Specifically, let us consider a family of systems 9
L , ( u ) e ( 1 -T) Lo(u)+ TL (u) = 0.
that depend on the parameter T E [ 0, 11, where Lo (u) = A U -4
(5.1)
421
EXISTENCE THEOREMS
Let us suppose that the following conditions are satisfied for (5.1):
a
for x E and arbitrary vectors u and p. For an arbitrary classical solution u(x, we have the inequality
T)
of the system (5.1),
(5.3) To see this, let u s take the scalar product of (5.1) and the vector 2u and let u s write the result in the form of an equation for 'u = I u (x, r ) 12:
Tf v ( x ) attains its maximum at any interior point x,, E 52, then the f i r s t t e r m in (5.4) is nonpositive at that point; that is, Tbl dv/dxl is equal to 0, and, consequently, this last impression is nonnegative; that is , (1 - r ) V - - b U d O .
F r o m this inequality and (5.2), we have (1 -r ) v + zclv - rc2 ,< 0.
Therefore,
This proves (5.3). After we have found a bound for max J u( x . ?b
r)l,
we can obtain all
subsequent bounds for u ( x , 7) under the conditions stated in Sections 1-4 if we note that these conditions [inequalities (2.1)-(2.4)] will also be valid for the entire family L,(u), where T E [0, 11, with positive constants independent of 7. In particular, these constants give a bound from above for max lVul for all possible solutions 9
u(x, r ) of the system (5.4) that vanish on S:
4 22
QUASILINEAR S Y S T E M S
max I Vu (x, Q
T)I
MI,
5
E 10, 1I.
(5.5)
All this enables u s to make the following assertion: Theorem 5.1. Suppose that inequalities (5.2) hold f o r x E G and arbitrary u and p. Suppose that inequalities (2.1) (2,4), where E ( M )
-
satisfies condition (2,8),are valid for x E s, for I u I 4 M, where LM is as in (5.31, and for arbitrary p. Su#pose that the functions bl (x, u, p) and b ( x , u, p) are continuous in the sense of Holder with exponent p on the set 9Jl( x E H, I U I
sM .
IPI
,< Mll.
.
where M , is the bound given in (5.51, and that S E C2,". -Then, if uIs = 0 , the system (0.3) has at least one solution u E C2, (9). I f , in addition, the a i j , the b , , and b on IIE below to the classes Ck,p(IIE), where k > 1, and i f S .$ Ck+2,B , then the solutions u will belong to ' k t2, p ('>*
This theorem is proven just a s in the case of a single equation with the aid of the Leray-Schauder theorem and the inequalities that we have obtained above. For T = 0, the system (5.1) is known to have a unique solution. We might state more general existence theorems [without the assumptions (5.3)] analogous to those in Section 8 of Chapter 4 by using the inequalities obtained in Sections 1-4. But we hope that the reader will not find it difficult to do this himself.
9 Other Devices for Obtaining Bounds for the Holder Constants for Solutions and Their Derivatives
In the p r e s e n t chapter, we shall, first ofall, give another method of obtaining bounds for the Hijlder constants for solutions (and t h e i r derivatives) of l i n e a r and quasilinear equations withdivergent principal p a r t and for the derivatives of the solutions of general quasilinear equations under the s a m e minimal assumptions regarding the functions generating the equations as in the preceding chapters F u r t h e r m o r e , we shall combine two difficulties: (1)unbounded singularities with r e s p e c t to x of the coefficients in the case of l i n e a r equations; and (2) nonlinearities (with r e s p e c t to u and p ) i n the case of quasilinear equations with principal p a r t in divergence form. Specifically, w e shall consider equations of the form
.
w h e r e the functions a f ( x , u , p ) and a ( x , u , p ) have singularities not only when u and p are equal to 00 but also when x 52. F o r the Q P%fori bounds in question to be possible, it is necessary that t h e singularities of the functions a, and a with r e s p e c t to x , u, and p be consistent with each other. The general tendency is as follows: T h e s t r o n g e r the nonlinearity, the weaker m u s t be the singularities with r e s p e c t to x . The equations considered in Chapters 3 and 4 r e p r e s e n t two e x t r e m e cases: (1) the a, ( x , u , p ) and a ( x , u , p ) are l i n e a r with r e s p e c t to u and p and have the maximum possible 423
424
OTHER DEVICES
F O R OBTAINING BOUNDS
singularities with respect to x ; and (2) thea, ( x , u , p ) and a ( x , u , P ) are bounded with respect to x and have maximum growth with resepct to u and p . Two special (limiting) cases of the results that we shall prove here regarding bounds on the Hillder constants are the corresponding results obtained in Chapters 3 and 4. We shall also give just such a generalization for quasilinear equations of the general form. A s we indicated above, all these generalizations are also possible within the framework of the basic method expounded i n the preceding chapters, which rests on the properties of functions in the classes 23,. The method proposed in the present chapter for finding a bound for the Holder constant is somewhat simpler and more common than the basic one. A l l the analytic facts comprising it are contained in Sections 1-5 of Chapter 2. However, the simplification i s achieved at the price of repeated reference to the equation. In the basic method, we use the equation only once, deriving inequalities (6.1) of Chapter 2 from it, these constituting the basis of the definition of the classes 3,. When this i s done, further study has to do with arbitrary functions satisfying these inequalities and not just solutions of elliptic equations. Therefore, the results that are obtained in this process are of more general significance than the derivation of a Priori bounds for the solutions of elliptic equations. In the method that we present, we derive two sets of inequalities from the equation, one of these for the solution u ( x ) itself (or its derivative); the other i s a set of inequalities analogous to inequalities (6.1) of Chapter 2 and applying to certain specially chosen convex functions v = 'p ( u ) , the boundedness of which ensures continuity of u ( x ) in the sense of Holder. (The idea of introducing such functions belongs to Moser [ 631.) The derivation of these inequalities i s somewhat more complicated than the derivation of inequalities (6.1) of Chapter 2. On the other hand, it is undoubtedly easier to obtain the necessary consequences of them (regarding the boundedness of the functions appearing in the inequalities) than to prove the Hillderness of functions i n the classes 93,. We shall first illustrate this method with the example of the very simple elliptic equation of the type
and then apply it to the general case. Here, we shall confine ourselves to interior bounds. Bounds close to the boundary can be found by the reader himself. A l l the analytic propositions that w e need are given i n Chapter 2. In addition, we shall give some other esthetic methods for finding bounds for the Holder constants for solutions of the different
THE CASE OF THE S I M P L E S T EQUATION
425
classes of elliptic equations, namely, Morrey's method [36] for two-dimensional, calculus-of-variations problems, Nirenberg's method [28] of finding a bound for the Hblder constant for the derivatives uxi of the solutions of two-dimensional, quasilinear equations of the general form, and Moser's method [ 631 of bounding where V c P , for solutions of Eqs. the Holder norm of I u (1.1).
1. THE CASE OF THE SIMPLEST E Q U A T I O N
Suppose that
11
( x ) satisfies
the identity
where 7 ( x ) E $1 (Q). Let us assume that
and that u ( x ) E W:(Q)and essential max e
I u I = M < 00.
We know
(see Lemma 4.8, Chapter 2) that to obtain a bound for 1 u I(, , e , where it will be sufficient, for example, to show that, for any two concentric spheres KR c U 2 R ~ ,Q P'cP.
osc [us
KA)
,< 8 osc
[ u p
K2R)
(1-3)
where the constant 8 is less that 1 and is independent of both u ( x ) and R . Without loss of generality, w e shall assume that the oscillation of u ( x ) i n KZRis equal to 1 and that 0 ,< u ( x ) ,< 1. A t least one of the following two inequalities holds for u ( x ) in K R :
Let us suppose, for example, that the first of these i s true. Then, we can carry out all our subsequent reasoning in connection with the function u ( x ) and use only the fact that it satisfies the inequality
4 26
OTHER DEVICES FOR OBTAINING BOUNDS
with a r b i t r a r y nonnegative 7 ( x ) E U% (KzR). In the opposite case, we would consider the function w ( x ) = 1 u ( x ) and u s e it instead of u ( x ) i n inequality (1.4) with yi ( x ) , still nonnegative. L e t u s define w ( x ) = + ( u ( x ) ) , where ~)(u)=-In2f~l-u+~). Here, E denotes a positive number that we shall let approach 0. If we c a n show that w ( x ) is bounded f r o m above in K R by s o m e constant M , independent of E , then we shall have the following inequalities in K R :
-
2(1 -u+t)>/e--Mn,
and, consequently, u(x)
.<1 --21
that is, (1.3) will hold with 8 bounded, let u s set
e-Ml,
= 1
- e-Ma 2 . To
show that w ( x ) is
in (1.4), where E(x) is a function that is of compact support in This yields
KZR.
or, what amounts to the same thing,
F o r E , let u s take the function C2 (I .Y 1 for
-
-yn
Ik), w h e r e C (7) is equal
T E [0, and d e c r e a s e s linearly in [ 3/2, 21 at z e r o at T On the basis of assumptions (1.2), w e conclude f r o m this that
=
to 2.
On the other hand, we know that u ( x ) is bounded f r o m below by the number - I n 2(1 + E) and is nonpositive on a set the m e a s u r e of which is not less than (1/2) mes K R . This, together with (1.6) i m p l i e s the inequality [cf. (3.6) in Chapter 21
427
THE CASE OF THE S I M P L E S T EQUATION
(1.7)
where c , is a constant determined only by Let us now set, in (1.5),
v , 1,and
n.
( v ( x ) - k k ; 0).
E(x)=C2(xjinax
where k is an arbitrary number and C(w) is a smooth function of compact support taking values in [0, 11 on the sphere Kp ,where p [ R , 3 R ] , and let us discard the first nonnegative term. This yields
J [ a i j v x i v x l P + a i j v x , (v -k) 2 ~ c . dr x~ ,<0.
(1.8)
Akt p
where A k , is the set of points x in K p at which w ( x ) > k . By using (1.2), we obtain from (1.8) (1.9) Ak. p
’k. p
A s we know (cf. Lemma 5.4 of Chapter 2), it follows from this inequality and from (1.7) that
essential max KR
v(x)
< M,,
where M , is determined only by v , 1, and n. Thus, we prove the desired assertion. W e can modify the end of this proof as follows: Instead of using Lemma 5.4 of Chapter 2 regarding the boundedness of functions v that satisfy inequalities (1.9), we can use the simpler Lemma 5.2 of Chapter 2. But, to do this, we need to use the elegant though rather difficult-to-prove John-Nirenberg lemma [ 851, which states that inequalities of the type (1.6), (1.7) for arbitrary spheres implies s - s u m a b i l i t y of v ( x ) over h‘3K2fo r arbitrary S:
J I v 1” d x ,< c,R”.
(1.10)
K3
T R
Suppose that we know (1.10) to be true. Let u s show that from (1.10) and (1.5) with E>/ 0, we can find a upper bound for v ( x ) in
4 28
OTHER DEVICES F O R OBTAINING BOUNDS
K R by using L e m m a 5.2 of Chapter 2. Without loss of generality, w e can a s s u m e that R = 1. In (1.5), let u s set
max ( w ( x ) - k k ; 0).
E(x)=P(x)
.
w h e r e w ( x ) = w ( x ) C * ( x ) and C ( x ) is a smooth function of compact support with range [0, 11 on the s p h e r e Ka,, This yields
J [a,
jvxjvx, c2 (w
-k )
+
aijv.r,wxic2
+
Ak
U fj"xj2rCxi
+
(w - k ) ] d x
,< 0,
(1.11)
w h e r e A, is the set of points in the s p h e r e Kah at which w ( x ) > k L e t US r e p r e s e n t the term atjW.r,'te'ziC2inthe f o r m
> 0.
and let u s w r i t e (1.11)as follows:
J [a,jvx,vx,P (w -k ) +
'k
jwxiwx,l
dx
Q
J [ 2 a i j w x , w ~ ~x , 2 a i j v x , ~ ~ (w x , - k ) ]d x .
Q
(1.12)
'k
Then, by using Cauchy's inequality (1.2) of Chapter 2, we obtain 1,
,I [+
\<
+
aijwx,w.rj
+ai
'k
+
2aije2~2~x~~x,
jvxiv.c,V
(w -k )
+
ai jG,C.xj
(w - k>]dx.
By collecting similar terms, w e get
J ai
jwxiwxj
dx
< 2 J a,
Ak
jcxicx,
( 2 ~ 2 ~ 2 V+ C ~ k )dx.
(1.13)
'k
For C ( x ) , let u s take a function that is equal to 1 in K1 , with I VC I ,< 2. On the basis of condition (1.2), it follows f r o m (1.13) that, f o r such
c (x)
9
v J I V ~ / ~ d x , < 8J p (2v2+w)dx. Ak
'k
T h i s inequality and inequality ( l . l O ) , which is valid for a r b i t a r y s > 0, e n s u r e validity of the inequalities
JI 'k
Vw P d x
,< c
('[
w'dx)
21s
mes
2 1 --
A,
,< c, rnes1-2/s A,,
for k
> 1,
(1.14)
429
B O U N D S ON HOLDER C O N S T A N T S
s > 0. I t i s sufficient for us to know that we can take in (1.14). For such s, it follows from (1.14), as Lemma 5.2 of Chapter 2 asserts, that w ( x ) is bounded from above in the sphere Kv,. This in turn means that u ( x ) i s bounded in K1 Both the procedures that we have just expounded a r e also applicable to the general case. The reader will see how to do this in the following sections, where we shall give the f i r s t of them since it is somewhat simpler and shorter.
for arbitrary s
>n
.
2. BOUNDS ON HOLDER CONSTANTS FOR SOLUTIONS OF EQUATIONS (LINEAR AND QUASILINEAR) WITH PRINCIPAL PART I N DIVERGENCE FORM Let us look at elliptic equations of the form
under assumptions regarding the functions a , ( x . u , p ) and a ( x . u . p ) that take c a r e simultaneously of the case of linear equations with unbounded coefficients and the case of quasilinear equations with maximum possible growth i n the functions a , and a with respect to p, that is, both the casesstudiedin Chapters 3 and 4. We assume that we already know the quantity M = max I u 1. W e assume that the al ( x , u , p ) and a ( % , u , p ) p E (- 00, 00) and that they a, ( x , u . p ) p , IlIIai(x. P)I i
where m
la(x,
51
a r e measurable for x E &, satisfy the inequalities
1111
>v I p Im -'po(x), v =const > 0. < I * I P I ~ - ' + ' ~ ~ ( X p=const, ).
P)I
> 1, where the 'pl ( x ) a r e nonnegative, and where II'po. 'P2IILglrn(9)' ll'plllLq,(m-l)(qs PI 4 > n.
In the case of linear equations
a
(ai/
"/
+ + +b, +a (4 +f (4=0
+ a , ( x ) ~ fl(x>)
u
(XI UXl
-
conditions (2.2) (2.5) can be expressed i n the form v>
0,
M , and (2.2) (2.3) (204) (2.5)
4 30
OTHER DEVICES
F O R OBTAINING BOUNDS
W e shall concern o u r s e l v e s p r i m a r i l y with i n t e r i o r bounds. To get a bound f r o m above for the quantity w h e r e Q' c 0, it will, i n accordance with L e m m a 4.8 of Chapter 2 , be sufficient to show that, f o r a r b i t r a r y K R contained (together with the s p h e r e K2R concentric with it) i n 0,
+R'
osc ( u s KR] ,< 8 osc ( u s K ~ R ]
(2 -6)
with constants r > 0 and B < 1. Here and below, all the constants will be determined only by n, M = mzx I u 1, and the numbers Y , p, rn, and q in conditions (2.2)-(2.5). We shall a s s u m e that the r a d i u s H
<
03
and that u ( x ) satisfies the identity (2.7)
f o r an a r b i t r a r y bounded function 7 ( x ) i n i l(Q). W e need only consider the case m n s i n c e the HGlderness of u ( x ) and a bound for I u *, follow immediately f r o m the injection t h e o r e m s (cf. Theorem 2.1 of Chapter 2) when m > n. On the basis of (2.2) (2.5), itfollows f r o m (2.7) that f o r bounded
<
rl(x) i n
-
i', (8)s
J a, ( x . fl* ux)7)21dx s eJ IP I v u Im+cpzl171 Id x . (2.8) L e t u s show that, under conditions (2.2) - (2.5), itfollows f r o m (2.8) with q ( x ) > 0 that e
w h e r e C(x) is a smooth function of compact s u p p o r t w i t h r a n g e [ 0 , 11 on function KZR.
BOUNDS ON H6LDER CONSTANTS
431
L ( x ) in In (2.8), let us set ~ ( x ) = e C ( x ) , where h i s a sufficiently large number, which we shall select later:
On the basis of (2.2)
- (2.4), it follows from this that
The last integral on the right can be bounded with the aid of Holder's inequality as follows:
(2.13)
To obtain a bound for the integral
we use Young's inequality in the following form:
432
OTHER DEVICES FOR OBTAINING BOUNDS
w h e r e E i s a n a r b i t r a r y positive number which, f o r the derivation of (2.9), we can set equal tol. When we substitute these inequalities into (2.11), w e obtain, a f t e r grouping similar terms, ( i v - p - (rn - 1) p)
4 eAM
[
J eAu1 V U lrn cnrd x ,<
K2R
J 1 vc l"dx+ (x+1) p mes1-mKzR+
p
K2R
The last term on the right can be bounded by u s e of Young's inequality as follows: m-1
L e t u s choose X in such a way that the coefficient to the integral on the left s i d e of (2.16) will be positive, for example, equal to v
(here, X = 1 + p m
v).
Then, f r o m (2.16) and (2.17), we have
v JehIvuImCmdxg K2R
[
m
4 e*"mesKzR 2 p m a x I V ~ 1 m + p ( ~ + m m ) e s - 7 KzR]. KZR
(2.18)
F r o m this, we see the validity of (2.9). The somewhat c r u d e r inequality (2.1 9) (which, a s is clear f r o m the derivation given f r o m (2.9), is valid for q > n ) , will be sufficient for o u r purposes. The constant c is determined only by rn. Ad, v , and p in (2.2) (2.5). Thus, we have proven Lemma 2.1. Suppose that the function u ( x ) E W:,l(KZR)(where 1 < rn 4 n ) , that
-
essential maxl u I = M K2R
433
BOUNDS ON H ~ L D E RCONSTANTS
that u ( x ) satisfies inequalities (2.8) f o r arbitrary bounded non( K 2 R ) , that inequalities (2.2) - (2.5) with q > n negative 7 ( x ) in G!,, are satisfied, where the n o m s of the y , in (2.5) are evaluated over the sphere K ~ Then, ~ . inequalities (2.9) and (2.19), where t(x) is an arbitrary smooth function of compact support taking values in [0, 11 on K P R , are valid foru ( x ) . Let u s now set about proving (2.6).Supposethat osc ( u . K Z R J = o.
L e t u s a s s u m e without loss of generality that 0 < u (x) w in K Z R . (If inequalities- (2.8) are valid f o r u ( x ) , analogous inequalities with the functions a,( x , w , w x ) = u l( x , v - c, v,) [satisfying the s a m e inequalities (2.2) (2.5) as d o the a, ] are valid for w ( x ) = u ( x ) + c; we can also r eg ard the function v ( x ) = u ( x ) + c a s a solution of a n equation of the type (2.1):
-
d
(ai( x . dx1
v
-c,
9,))
+ a (x, w -c, w,) = 0. )
L e t us prove l e m m a 2.2. Suppose that x
EK R ,
u ( x ) ,< rnax u ( x ) K2R
- 8,w=(1
-82)
>(1 -
o
)>
(2.2 0 )
83) meS K R ,
where 8, and 8, are any two positive numbers. Suppose also that the assumptions of Lemma 2.1 are satisfied with q > n. Then,
-
osc { u , K R } ,< rnax u ( x ) ,< (1 KR
-6,) o +R',
(2.2 1)
where r = 1 7' with 8, > 0 determined only by 6, , 6, , rn, M , V , cc , and q in (2.2) - (2.5). Proof: L e t u s consider the function w ( x ) = $ (u ( x ) ) in K Z R ,where +(u)=-In
w-uU+R' 820
Thi s function satisfies the inequality w ( x ) > -In( la2). If we can show that the function w ( x ) is bounded fro m above, for a r b i t r a r y x E K R by some number M , independent of R , that is, if -In
o--u(x)+ 820
R'
S MI
everywhere in KR,we shall then have w-u(x)+R' 820
which is inequality (2.21) with a, =
' <
eMl.
a2e-MI.
,
434
O T H E R DEVICES F O R OBTAINING B O U N D S
.
Thus, our goal is to find a bound from above for v ( x ) in KR We shall do this with the aid of Lemma 5.4 of Chapter 2. But to be able to use this lemma, we need to have a bound for the norm 11 11 L, (Ka,%R)' Let us show first that ~ ( x )satisfies the inequality (2.22) where the constant c depends only on m , M , q , v ,and M . This inequality is analogous to inequality (2.19) for u ( x ) . Its derivation differs somewhat from the derivation of (2.19) because here we do not know the value of max IvI. On the other hand, we do have inequality (2.19), which we shall use in what follows. In (2.8), let us set
where C ( x ) is a smooth function of compact support with range [0, 11 on K 2 R . This is possible because ~ ( x possesses ) all the required properties (in particular, 71 ( x ) >/ 0). This gives us
-
On the basis of inequalities (2.2) (2.4), this last inequality and the definition of Y yield
W e find bounds for the integrals on the right in much t h e same way as we did for the integrals Ji above [cf. (2.12)-(2.15). Specifically, by using H6lder's inequality, we get J; =
J cqoCmB-md x 4
K2R
4 35
BOUNDS ON HOLDER CONSTANTS
Then, by using Young's inequality, w e have, for a r b i t r a r y
Jk,/ (e m I vw
rn lmcmE"-'+-~,
rn
VClmE-nt)
dx.
E>
0,
(2.27)
K2R
Again with the aid of Young's inequality with for J; =
s
I Vu 1'" B'-"'Cm d x =
sI
E
> 0, we find a bound
Vv
("*-' C"-'
+ , 1
I V u 1"
1 V u I CdK.
K2R
K2R
Specifically, we have
I Vv Im
Cme3
C m ~ - m ) dx.
(2.28)
-
L e t u s substitute these expressions (2.24) (2.28) into (2.23) and group s i m i l a r terms, assuming E ,< 1 and R ,< 1:
If we choose E in such a waythat the coefficient on the left is equal, f o r example, to ( m l)v/2 and u s e inequality (2.19), we obtain (2.22) from (2.29).
-
436
OTHER DEVICES F O R OBTAINING BOUNDS
From inequality (2.22), the boundedness of v ( x ) from below [specifically, v ( x ) > -In($ )], and the fact that, by virtue of the assumption (2.20), v ( x j is' nonpositive on a set whose measure is not less that (1 a3 ) mes K,, it follows that
-
(2.30)
with constant c, determined only by m y M , v , I.L,6, ,and 6, (cf. inequality (3.5) of Chapter 2). Let us show that v ( x )satisfies the following series of inequalities:
r
JIVvlm:'"dx P
(v-k)'"IVClrndx+
[,$
+p-'rnmes
1
1-5
.
Q
(2.31)
A,,,p
for all k exceeding some number k , and p < p,, Here, PI,<, is the set of all points x in the sphere K , at which u ( x ) > k and i ( x ) is a smooth function of compact support with range [0, 11 on K p The constant r is determined only by m , M , pn , v ,and I.L ,and the constant k , is determined by the constants a2 , /Myand V . For verification of (2.21), w e need only prove inequality (2.31) for the spheres Up concentric with K Z R , where p E [ H , 3K] and R 4 1. To prove (2.31), let us take, i n (2.8),
-
3 ( x ) = C'" ( x ) Bl-'"
IlldY ( V
.
( x ) - k , 01,
where B ( x ) = cu u ( x ) + R' and C(x) is as before. Such ~ missible. We write the result i n the form
J[ui5'-'%rL i'" +(rn - 1) u , 5 - " ' u r'(v - k ) C"']
dx
Then, keeping (2.2) v
J [l
ad-
<
(2.32)
p
+ J (p I v u I '"+(p,)
( x is )
5' -nr (v - k ) C"' d x .
Ak, p
- (2.4)
i n mind, w e obtain
~ v l ~ (m~ ~ 111vvlm + (v - k ) C"I d x
,<
Ak. p
4
f[(poB-"'c'"+(m-
4
P
f r n (p I v u
+
(p])
l)cpo5-"1(v-k)C'"-+
B'-"(v - k ) :'"-I
I VCI
+
(2233)
437
BOUNDS ON HOLDER CONSTANTS
We find bounds for the integrals on the right in essentially the same way a s we did above for the integrals .I, and J;. A slight modification is necessary because of the factor (v - k ) Specifically, applying Holder's inequality, we have
.
& = J 'poB-"Crn11 +(m- 1) (9- k ) ]d x ,< 'k.
o
L
rn
W e find a bound for the factor(I(v-&)C(I Lrn ('k, equality (2.12) of Chapter 2:
p)
with the aid of in-
Then we find a bound for the term containing it by using Young's inequality with E > 0:
(2.34)
To get a bound for the integral
438
OTHER DEVICES F O R OBTAINING BOUNDS
we use th? inequalities just derived. In comparison w i t h d , the integral J1 has only the "extra" factor 8 = w u + R'. Since this term is bounded, we can take it outside the brackets, so that, from (2.34), we get
-
52"4 (w
+R')[c,R-'"
mes' - ;Ak,
Clem
p+
+
c1
II IVwl ell;m
Il(v -k ) I VCl
+
(2.35)
I.",].
In almost the same way, we find a bound for J;:
I VCJ d x <
vlB1-"(W - k ) C"-'
J; = Akv p
< R-r(rn-l) J '91 (y -k ) IVCl d x 4
\
Ak,
D
From this, we obtain, by using Young's inequality,
=
'k, p
f
Ivwl"-'
(w-kk)Cm-'pC(
dx
Ak, p
just as we did above for J i by "joining"
Jl ,<
j - (+ pw,"C",,-1+--(w "
-k)m
1
to IVCl:
(w - k )
)VCI"e-"
Ah p
1
dx.
(2.37)
W e still need to find a bound for the integral J," =
1 VU 1"
(V
Ak. p
F o r all
w
s
- k ) CmB1-md x = (I V v l C)"
(V
-k ) B d x .
Ak. p
E Ak, , the function @ ( x . k , w) = [w ( x ) -k ] B ( x ) Iw ( x ) - kj 8 2 0 e - v ( x )
,< 2M and
x
L-
does not exceed the function as k +m. Therefore,
@ (k)= 2 M 6 , e - 1 - k ,
which approaches 0 (2.38)
439
BOUNDS ON HOLDER CONSTANTS
If we substitute inequalities (2.34)-(2.38) into (2.33) and collect similar terms, we obtain (remembering that R 4 1 and r > 0 ) ~
cl~'"-p(m--l)
(v-(2M+2)
rn
~
-
s
(pk ) ) @
Vv 1"'
C"'
dx+
p
+v
~(m-I)~Vv~"'(~-k)C"'dx,< P
Let u s choose
E,
for example, in such a way that
(2M
+2 )
+ p (m- 1)
C , E ~
m
=
Let u s assume that the numbers k exceed a number k,, this latter chosen from tine condition
>
A s one can easily see, for k k , and R ,< p ,< 3/,R ,< y2, inequalities (2.31) follow from (2.39). Thus, inequalities (2.30) and (2.31) are established for the func-
tion v under the assumption that r = 1 -
4
> 0 (that is, that q >n)
and
that k 2 k,,, R ,< p 3/,R ,< Y2. The constants in these inequalities are independent of R , 2, and They are determined only by m,M, V , p, 8,, and a,. On the basis of Lemma 5.4 of Chapter 2, it follows from (2.30) and (2.31) that,forq > n,thequantitymaxv(x) is bounded from above (I).
KR
by a constant M,determinedonlyby c,in (2.30), by in (2.31), and by the numbers k,, n,m, and y, that is, in the final analysis, by n, m , M, q, v , p, 6, and 6,. This completes the proof of Lemma 2.2. It follows from this lemma that if u ( x ) satisfies inequalities (2.8) for arbitrary bounded ~ ( xin) W k (KZR),it satisfies inequality (2.6). To see this, let us assume, just a s above, that the function u ( x ) varies in the interval 0 ,< u ( x ) ,< w for x E KYR,Let us consider the two functions a',( x ) = u ( x ) and w, ( x ) = w - u ( x ) . At least one of these satisfies the inequality
440
O T H E R DEVICES F O R OBTAINING BOUNDS
If (2.41) is valid for I = 1, then Lemma 2.2 ensures the validity of inequality (2.6) for 7 = 1 -3, and 6, = 6, = 1/2. If (2.41) is valid for I = 2, then we can apply Lemma 2.2 to the function w2. Specifically, condition (2.20) is satisfied for this function 1/2. Of the other assumptions of the in place of u ( x ) with 6,=6,= lemma, we need only show that w 2 ( x ) satisfies inequalities (2.8) with q ( x ) 0. But we know that this is the case because of the validity of inequality (2.8) for u ( x ) , where q(x) can be of aribtrary sign, that the following inequalities hold for w2(x) :
>
A s one can easily see, in these inequalities, the functions
-
a , ( x , w . p ) =- a , ( x .
w
-w , - p )
satisfy the same conditions (2.2)-(2.5) a s do the functions a , ( x , u . p ) . Thus, Lemma 2.2 can be applied to w - u ( x ) . Just a s in the first case above, this leads to inequality (2.6) with the same 7 = 1 -6,. Let u s formulate this assertion a s lemma
2.3. Suppose that
u (x)E
W!,,(K2,d (where 1 < m Sn),that
essential max I u 1 = M K2R
that the function
u(x)
satisfies inequalities (2.8) for an arbitrary
boundedfunction ? ( x ) E &,(K2,+ that thea, ( x . I(, p ) andv2 satisfy conditions (2.2), (2.3),and (2.5) wathq > n , where the n o m s of the pi in (2.5) a r e evaluated over the sphere K Z R .Then, for some 6,>0,
Here, the constant 6, i s determined only by n, m , M , 9 , v , and 1 in (2.2), (2.3), and (2.5). Instead of the requirement made regarding the validity of (2.8) a s applied to u (%)forarbitrary ql(x), we can assume that, for each of the two functions w ( x ) = k u ( x ) , an inequality of the form
44 1
BOUNDS ON H ~ L D E RCONSTANTS
holds. Here, the functions a? ( x , w , p ) and y2 are assumed to satisfy conditions (2.2), (2.3), and (2.5), and ~ ( xis) a nonnegative bounded element of i k ( ~ ~ ~ 1 . From this lemma and Lemma 4.8 of Chapter 2, we get Theorem 2.1.
Suppose that
u ( x ) E W i t 69,where 1
essential max e
< m ,< r ,
that
I u ( =M
that inequalities (2.43), in which the functions a t ( x . w. p ) satisfy conditions (2.4, (2.3), and (2.5) are valid f o r the functions w (x) = f u ( x ) , and that 9 (x) i s an arbitrary nonnegative bounded function in @k(8). Then, f o r an arbitrary sphere K , c O , osc ( u .
KP),< cp,ap=.
(2.44)
Here, c and 3 are positive constantsdetemined only by M , n, m, q, v , and p, and po is the distance f r o m the center of K, to the boundary of the region 51. In particular, the assertion of the theorem is valid for bounded generalized solutions of Eq. (2.1) belonging to Wf,(Q) that is, for functions u ( x ) E W', ( Q ) such that essential max Q
J u I= M
< 30
that satisfy the identity (2.7) for all bounded ~ ( xin) i ' , ( S ) if conditions (2.2)-(2.5) are satisfied for the a i ( x . u , p ) and a ( x , u , p ) . A bound for J U ( ( ~ ) , for the entire region 9 is ensured by Theorem 2.2. Suppose that the conditions of Theorem 2.1 are satisfied, that S satisfies condition ( A ) , that u Is = Q ( s ) satisfies a HCilder condition with exponent (3 > 0. Then, there exist constants > 0 (with LY< 8) and c > 0 that depend only on M, n, rn, 9 , V , I.L,and ?, on the constants a, and 0 , in condition ( A ) ,and on I u lo, s, such that (2.45)
This theorem is proven in the same way a s Theorem 2.1 was proven. The reader can find the changes that have to be made in the line of reasoning in Section 14 of Chapter 3 and in Section 1 of Chapter 4.
442
O T H E R DEVICES F O R OBTAINING B O U N D S
3. BOUNDS ON THE OSCILLATIONS OF THE DERIVATIVES OF SOLUTIONS OF EQUATIONS WITH PRINCIPAL PART IN DIVERGENCE FORM If u ( x ) satisfies Eq. (2.1), we can look at each of i t s derivatives u k ( n ) = u (xj as the solution of an equation of the same type. Xk
Specifically, let u s differentiate (2.1) with respect to write the result in the form
xk,
and let u s
This equation is alinear equation (inthe i d k ) of the form (2.1) with a , (x.
'u.
p ) = u i, ( x ) P i
a(x.
'u,
p)=o,
+if v + (x)
fi
(4
where
In view of this, from Theorem 2.1, we get
Theorem 3.1. Suppose that u ( x ) ~ W ; ( & )that , essentialmax1Vul= and that u ( x ) satisfies E q . (2.1) almost everywhere. Suppose that the functions defined in (3.2) satisfy the inequalities M,
< 00
Then, f o r arbitrary Upc Q ,
<
osc phi.UP) cp;"p".
(3.4)
where PO is the distance f r o m the center of K, to the boundary S,and c and a a r e positive constants determined only by M ,, n, and the constants q. V, and p in (3.3).
E Q U A T I O N S NOT IN DIVERGENCE F O R M
443
4. EQUATIONS NOT IN DIVERGENCE FORM
In the present section, we shall consider quasilinear, secondorder equations of the general form L u r a , j ( x , u,
U X ) U X1. , J
+ a ( x . , u , u,)=O
(4.1)
in a region Q and we shall find a bound for the corresponding Halder constants I uxl p , , where 1 = 1, . . ., n , in terms of max I u I = M , and P ",ax lVul= M,, the constants characterizing the functions a i J ( x ,u , p ) and a ( x . u . p ) . and the distance from 51' to the boundary S. The only assumption regarding (4.1) (outside of the assumption of some regularity of aijanda) is that it is elliptic at the solution u ( x ) that we are considering. We proved this result in Section 1 of Chapter 6. Here, we adopt the general plan used in that section for obtaining bounds for 1 u,. 1 but we modify the end of it by replacing proof of HiSlder1 C),s!' ness of the vector-valued functions belonging to t.he classes B? with somewhat more usual and briefer considerations similar to those of the preceding sections. Thus, let u s prove a theorem that is a slight strengthening of Theorem 1.1 of Chapter 6. Theorem 4.1. Suppose that u(x) is a solution of Eq. (4.1) belonging to w ~ ( QSuppose ). that essential max I V u l g M , . P
Suppose that the functions a,,(x, u , p ) are differentiable with respect to n, u , and p in a neighborhood of the manifold ( x E s l , u=u(x), p=u,(x))
and that
vz E : . < a i j ( x . n
i=l
Suppose, finally, that for
n
u ( x ) , u x ( x ) ) E i E J < p z E:. i=l v, p = const 0.
>
u = u ( x ) and p = U , ( x )
(4.2)
,
Then, u (x) belongs to Cl, (51) f o r some a > 0, and the norms I uxi la, *,, where 1 = 1 , , n , f o r an arbitrary subregion 51' c 51 are bounded in terms of n , M,, q , Y , and p f o r the distance from &' to the boundary of Q . The constant a i s detemnined only by n , A,,q , Y , and H.
...
'1
4 44
OTHER DEVICES F O R OBTAINING BOUNDS
w; ( x ) = 1 Onuxl ( x )
+2 n
i=l
u2 xi
(x), n
w:
10n(l -uxl(x))+
(XI=
2 u2xi ( X I ,
r=1
and for an arbitrary nonnegative function inequalities
J [ni ( x , ]
+
" : ]
P
2c: (XI]TXi d x
~ ( xin) W:
. . ., n,
(4.4)
(P),we have the
s
4 -J l [ I V"'12+
-
l = 1,
(4.5)
'2 (-41'1d x .
P
where a i j ( x ) = a i j ( x . u ( x ) . u X ( x ) ) and the norms ,e) and(Iy(ILg,2(s, 4 a r e bounded from above by constants determined, like 7 , by the numbers MI and cc in (4.3).The inequalities (4.5) are consequences of the identity (1.9), proven insection 1of Chapter 6, that is, of the identity
where
q ( x ) is
an arbitrary bounded function in
Gi (Q)
and
On the basis of the hypotheses of the theorem, the coefficients and c: a r e q-summableover Q and the all are bounded. Let u s
a;;. c i j
find bounds for the t e r m s in the integrandon the right side of (4.6), using inequality (1.2) of Chapter 2 a s follows:
Ia;uXmxi":, '1I s 4m i '1+ r2(":y '1. I c : p xi l '1I s I '1+ 1 (c: '1s s1 +r1 'p ( x ) '1. x
x
1
U X i X j 12
])2
E
UX'X,
12'1
I
EQUATIONS N O T I N DIVERGENCE F O R M
445
wh e r e the function y ( x ) h a s a finite norm II(PII~,,~~~,. that d o e s not exceed a quantity that we know. When we substitute these bounding expr e ssi ons into (4.6), choose E sufficiently small and collect similar t e r m s , we obtain inequalities (4.5) for nonnegative functions q ( x ) . These inequalities constitute a special case of inequalities (2.8). They coincide with (2.8), with nr = 2, and a f ( x . U. p ) = “ a , ( x , u ( x ) . u , ( x ) ) p , $-2 4 ( x . u ( x ) . u x ( x ) j .
The remaining assumptions regarding the functions ‘pi and a, also correspond to each other. However, th e r e is one fundamental difference: Inequalities (2.8) are valid for both the functions II ( x ) and -u ( x ) , w hereas inequalities (4.5) are valid only for werl(x) and not f o r - w f( x ) However, inequalities (4.5) are valid both f o r w: ( x ) and w ‘ _ ( x ) , and the role of the p a i r k u ( x ) is now played by the functions w: (x). A s we can see f r o m the considerations in Section 2, satisfaction of the inequalities f o r both the functions 1- u ( x ) w a s used in only one place, namely, when we’ made the choice between these two functions according as the inequality
.
{
mes x
E !
u (x)
> 1 mes K R
<};
o r the inequality
{
mes x
EKR. u (
> 91 rnes K ,
x ) > ~ }
w a s known to be satisfied. If, fo r example, we used the f i r s t inequality, then all the subsequent reasoning was c a r r i e d out with r e g a r d to the function + u p ) , and this inequality was used only to conclude that inequality (2.30) follows fro m inequality (2.22). In h e pr e s en t case, we shall proceed as follows: L e t u s take two a r b i t r a r y concentric s p h e r e s K R c K Z R C Q . Out of all the functions 1 1 , ~ ( x ) , where 1 = 1, , n, let u s choose the one that h a s the greatest oscillation in K Z R .Suppose that
...
~ O S C( U x , , K 2 R J I = W r =
inax 0‘. .... n
i=l.
In accordance with Lem ma 8.1 of Chapter 2, osc
p;. K Z R )>60(or
and the following inequalities will be satisfied f o r at least one of the functions w: : mes x
{
E K,;
w r ( x ),< inax 70‘KZR
8,pr
(4.9)
446
OTHER DEVICES F O R OBTAINING BOUNDS
where 6, , 6, , and 6, are certain completely determined positive numbers. Suppose, for example, that this is true for w: Then, all the hypotheses of Lemma 2.2 a r e satisfied for w : . Therefore,
.
for some 6, > 0. From this, weconcludeon the basis of Lemma 4.9 of Chapter 2, that u,, satisfies a Holder condition and that we can obtain a bound from above for I u, I ,where P' c Q. This completes 1 (a). 0' the proof of Theorem 4.1.
5. MOSER'S METHOD OF BOUNDING I I I I ~ , ~ , FOR SOLUTIONS OF LINEAR EQUATIONS In connection with equations of the form (5.1)
Moser proposed in [ 631 a method for finding a bound for IuI=, ?,, where ii denotes a generalized solution of such a n equationin
W;(Q)and 8' c Q , i n terms of inequality
I I U I I ~ , ( ~ ) , the numbers
v
and
p
in the
and the distance from 8' to the boundary S. H i s method is different from the prior methods of de Giorgi and Nash. Expanding that method further and using the John-Nirenberg theorem [85], he showed that, for solutions of Eqs. (5.1) belonging to Wi(-) 0 that are positive in L? and whose coefficients satisfy conditions (5.2), Harnack's inequality min u ( x ) 8'
> c inax u ( x ) , 9'
(5.3)
is valid, where the positive constant c depends only on n, v , p , and the distance from Q' to the boundary of the region 52 (cf. [70]). W e shall expound this method of finding a bound for 1 u ( , *,, assuming, for simplicity, that the solution u is bounded. Thus, let us suppose that u ( x ) belongs to W: (Q), that it satisfies the identity
M O S E R ' S M E T H O D FOR BOUNDING I U ,.I e,
447
(5.4)
for a r b i t r a r y q ( x ) E L% (Q),and that m L - 1111 < 00. O 1 ) pr ( U (x))E ( x ) , where E(x) E Wz(Q),and In (5.4), let u s set ~ ( x= let y ( t ) be a twice continuously differentiable function of t such that y " ( t ) > , 0. This yields
whe r e v(x) =
y(u(x)).
It follows fro m this that, f o r E(x)> 0,
(5.6) We shall refer to a function v ( x ) that satisfies inequality (5.6) f o r a n a r b i t r a r y nonnegative function E(x) in W:(Q) as a subsolution of (5.1). I t is e a sy to verify that the following inequality is valid for solutions u (x) and nonnegative subsolutions v ( x ) of Eq. (5.1):
where K , is a n a r b i t r a r y sp h ere belongingtoQ and C ( x ) is a smooth function of compact support with range [0, 11 on K , . To see this, let u s set 7 ( x ) = u ( x ) P ( x ) in (5.4) [resp. E ( x ) = v ( x ) C ' ( x ) in (5.6)l and c a r r y out the elementary bounding operations with the aid of Cauchy's inequality (1.2) in Chapter 2. L e t u s f i r s t find a bound fo r m,wlu (x)I, where K R c 8. W e recall KR
for E > 0, are nonnegative subsolutions of (5.1). T h i s is t r u e because the functions v6 = ( I u I +6)'.'-' are subsolutions for a r b i t r a r y 6 > 0. If we let 6 approach 0 in (5.6) fo r v 6 ,we see that inequalities (5.6) are also valid for v . Therefore, inequality (5.7) is valid for v ( x ) ; that is, for a r b i t r a r y E > 0, v ( x ) = Iu ( x ) l '
(5.8) In addition, we need the injection theorem (2.12) of Chapter 2 i n the following form: 1
448
OTHER DEVICES FOR OBTAINING BOUNDS
where k = -2and n-2
It is valid for a n a r b i t r a r y function W ( X ) in @(KP). L e t u s take a sequence of concentric s p h e r e s K h = K R h , where 1
Rh = R ( l +2;.) f o r h = 0, 1, 2 ,
... and K 2 , c 3, and let ,us take the
corresponding sequences of functions vil= I II l h h , and the corresponding functions CI, ( x ) such that c h (.Ic)
0 = in
= 1 = outside K,t
and
W e define
f v; d x = Oh. Kh
On the b a s i s of the definition of v / , and inequality (5.9), we have
L e t u s u s e (5.8) with 1 + E = h" and p = K h to find a bound f o r the right-hand m e m b e r of (5.10). This yields
that is, the r e c u r s i o n relation @h+l
,
h=0,
1, 2,
.. .,
(5.11)
M O S E R ' S M E T H O D FOR
where c = [ P 3 e ( % + Chapter 2) that
I)]'.
BOUNDING 1 U la,
0,
449
It follows from this (cf. Lemma 4.7 of kh-l
kh-l
h
0 .
and hence that
where c, = max { 1: c } Thus, we have proven Theorem 5.1. Inequality (5.12), with the constant c, depending only on V , IL, and n , i s valid for bounded generalized solutions u ( x ) in W:(9)of E q . (5.1). Remark 1 . The assumption regarding the a prio?"iboundedness of u can be discarded. Remark 2. It is easy to see by going through the derivation just given of inequality (5.12) that, if w ( x ) is a nonnegative subsolution of Eq. (5.1), then (5.13) with the same constant c1 as in (5.12). Let us now seek a bound for the Holder constant for a solution u ( x ) of Eq. (5.1). Let us take three concentric spheresKRcK,,c K3* belonging to 9. Suppose that w2 = essential rnax u ( x ) and that w, = essential min u ( x ) on the sphere KaR. At least one of the two inequalities
is always valid. If we can show that inequality (5.14) and Eq. (5.4) (5.4) for u ( x ) imply
(5.15)
4 50
OTHER DEVICES FOR OBTAINING BOUNDS
it will follow from (5.14,) and (5.4) that max[w,+w2-u(x)]
< w2-8w.
since (5.14, ) and (5.4) a r e now valid for the function u ( x ) = w, + I U ~ u ( x ) and hence so is (5.15). This proves that (5.4) implies
-
osc ( u , K H }4 (1 -8)osc
(u.
K3R].
(5.16)
This, on the basis of Lemma 4.8 of Chapter 2, enables u s to obtain a bound for Iu I,n),P,. Thus, let u s suppose that (5.14, ) is valid for u ( x ) . Let us consider the function
where 8, is a positive number, which we shall l e t approach 0.. This function is a positive subsolution of Eq. (5.1) since
Therefore, on the basis of Remarks 1 and 2 following Theorem 5.1, inequality (5.13) i s valid for it essential maxw ( x ) ,< c2 KR
(Ki
w 2dx)"
.
(5.17)
To get a bound for the integral on the right, let us write the identity (5.5) that w ( x ) must satisfy. It is of the form S ( a i j ( x ) w , ~ ~ w , ~ aijWer,.E.r.)dx=O. :+ I '
9
Here, let u s take E(x) = C2 :(t)=
1:
(Ix
(3H
(5.18)
- x"I), where
-f)/H
for t.< 2 K for 2R;( t 4 3 K for t > / 3R
and x" is the center of the sphere K 3 R . Then, on the basis of (5.2), it is easy to show from (5.18) that
MOSER'S METHOD FOR BOUNDING 1 U
On the o t h e r hand, at those points of the function w ( x )
+ 28 :In 1 1 1= I n (0)
I
T+b
K2R
1%.
45 1
9t
at which u ( x ) <
(w
),
2, and since the m e a s u r e of such
points is, by the assumption (5.14, ), no less than
1
mes
KzR,
in-
equality (3.5) of Chapter 2 yields
J
lw(x)-1n2j2dx < P ' R ~ J ~ ~ w l 2 d x .
2 , 2R
(5.20)
2. 2R
w h e r e p' is a constant depending only on n. Here AI, 2. 2R is the Set of points x in KzR at which w ( x ) > In 2. On the b a s i s of (5.19) and (5.20), w 2d x
< c3RN
K2R
(5.21)
F r o m this and (5.17), we obtain essential maxw ( x ) < c4. KR
(5.22)
Here, the constant c 4 , like all the other constants in o u r bounding expression, is determined only by n, V , and 1.1 and is independent of R , u ( x ) , and 6, > 0. If we let 6, approach 0, we see that oessential maxIn L w * - u ( x ) g c4, KR
that is, (5.15) holds with 6 = e-ca. This completes the proof of inequality (5.16). Moser's method differs from the method expounded in Section 1 in i t s proof of the boundedness of the auxiliary convex function v = = y ( u ) . Moser does this by obtaining r e c u r s i o n relation (5.11) f o r the integrals
F r o m this r e c u r s i o n relation, he concludes that 0 j h ,where / I = 1,2, , and hence essential max I'uI are uniformly bounded. Our
...
KR
procedure is to obtain inequalities (1.9) f o r 'u that contain a n a r b i t r a r y p a r a m e t e r k and to conclude from them that essential m a x 1v1 is bounded. T h e r e are two ways of investigating equations KR
452
O T H E R DEVICES F O R OBTAINING BOUNDS
of the general type ( O . l ) , using the basic idea of one method o r the other. We show in Sections 2-4 how this can be done by the f i r s t method. We can take the second method so that it will coincide with the first everywhere except as regards proof of the boundedness of v . To find the boundedness of J v l , we can obtain the recursion relation ah,, ch + (which is similar to (5.11) and inwhich c is aconstant determined by known parameters). From this inequality it is easy to conclude that IvI is bounded. This is ,easily done by using the relations and inequaliEies derived in Sections 2-4. However, all the transformations and bounding operations necessary by this method are somewhat more tedious than in the case of the first method, which is similar to that expounded in Sections 2-4.
6. NIRENBERG’S ESTIMATE We shall now expound a simple and esthetic method of bounding the HCilder constant of the first derivatives of solutions of quasilinear elliptic equations
z aij 2
LU1.
j-I
(.x,
This procedure was proposed by Nirenberg in [28]. I t is strictly 2-dimensional and cannot be carried over to the case, of equations with three o r more independent variables. Let u ( x ) be a solution of the equation. We assume, for simplicity, that u ( x ) is twice continuously differentiable. Suppose that 18 ( x i satisfies -the following inequalities :
The functions ail ( x . u ( x ) , u X ( x ) ) and n ( x , u ( x ) , u x ( x ) ) of the single variable x will be written simply a,, and a. Let us represent the expression
in the form
NIRENBERG'S ESTIMATE
453
On the basis of (6.1) and (6.2) and this representation of I, we obtain
Let K , c0. Let us assume that the coordinate origin is a t the center of this sphere, so that K,= ( r = 1x1 < p ] .
L e t C (x) = L( r) be any smooth function of compact support taking values-in [0, 11 on the sphere U p ,wherelVCl
.
In the second t e r m of the integral I2
.
=
J r-=C2 !u:,x, - Ux,x,ux,.rJd x
KP
let us integrate by parts once with respect to
x1 and once with respect to x 2 After collecting similar terms, we obtain
where
is the result of partial differntiation with respect to the angle 0 in a polar coordinate system r , 8. We find a bound for the second t e r m of the integral as follows:
4 54
OTHER DEVICES F O R OBTAINING B O U N D S
where E is an arbitrary positive number. W e represent the first t e r m preliminarily in the form I ----a
r' / r-3-1C2( r )-r1 -du, fl dO
J
4-
0
ux,r d r d8 =
0
which is possible since
and we use the inequality
This yields
0
2n
If we substitute the expressions we have found for1/,1 and//,[ into (6.6),we obtain lI,I
4
/.[(E+
u ~ t r j + E - l r - ~ ~ ~ 2 udx: ~ . ,4 ]
2x?a)r-"i2 i, j=l
lip
.< + 2 x 2 a )I, + (E
E-'
2x
c2M;
2--a p-",
(6.7)
NIRENBERG'S
455
ESTIMATE
where M, = inax 1 V I L1. KP
Now we substitute this expression into (6.5) and choose E and a sufficiently small that
__ V
2P
For such
E
and
a,
2 (E
+2 x 2 a ) > 0.
(6.8)
we have
that is, for arbitrary K -P and the spheres K p and K d concentric with it, 2 2
2
u2
St-.
K,-
"i xj
dx<
i, j = 1
Sr-"P Kd
u:,,,dx i, j = 1
11
(6.9)
2
for
This inequality (cf. remark following Lemmas 4.1 and 4.2 of Chapter 2) makes it possible to find a bound for the HiSlder ,where i = 1, 2, for arbitrary P ' c P . constant I u,! I ($), 9' For spheres K , that intersect the boundary, the general procedure of proof is the same: Wefinda bound for. the integral I , over K , n P. Here, we assume that we have upper bounds for the absolute values of the first and second derivatives of u with respect to arc length s of the boundary S. In integrating by parts i n I , , we single out terms along the intersection of the circle K , with the boundary S. To find bounds for these, it is desirable to straighten out the portion of the boundary in a neighborhood of which we are investigating u and to carry out all our reasoning in the new coordinate system. Let us suppose that this transformation has already been done and that the portion S, of the boundary S in which we are interested lies on the x, -axis. After integrating by parts, we get the integral p
I, =
r-n{2ux,uxlxl dx2.
sP
But this integral does not exceed the quantity
456
OTHER DEVICES F O R OBTAINING B O U N D S
which we know. W e encounter one further difference in finding a bound for the integral 14. It is due to the fact that, here, the integral
will not, in general, be taken over a closed contour and therefore will not necessairly vanish. However, we have bounds for its absolute value:
This information is sufficient for us to derive the desired bound for l1 according to the plan described above. Thus, we obtain a bound for I u x i ] ,where Q ’ c Q , for Z = 1,2,in terms of max I V u l ,
(4)
9’
9
the constants v and p in (6.2) and (6.3), and the distance from 9‘ to Q and we obtain a bound for I u x i ( in t e r m s of max IVul, the
(3)s
9
constants v and p in (6.2) and (6.3), and the norm in C, of the boundary values of u under the assumption that S cC,
.
7. MORREY’S METHOD FOR FINDING A BOUND FOR THE HOLDER CONSTANT FOR SOLUTIONS OF TWO-DIMENSIONAL VARATIONAL PROBLEMS Suppose that the integrand of the functional /(u)=j-
P ( x . u , u,)dx
(7.1)
9
satisfies the inequalities vlp124F(x.
where
v
P)
and p a r e positive numbers, for x E
(7.2)
n and arbitrary u and
p. Suppose that u (x) is a generalized solution in W: (Q) of the variational problem On the minimum of I ( u ) under some boundary condition such that u ( x ) minimizes the functional (7.1) out of all the functions W: (9)that satisfy this boundary condition (which, in particular, may be absent). Let u s show that u (x) will then be continuous in the sense of Hblder. W e confine ourselves to interior estimates.
457
M O R R E Y ' S METHOD FOR F I N D I N G A BOUND
L e t us take an arbitrary circle K R c Q and the circles K , concentric with it, where p < R. The function u p ( x ) that is equal to the function u ( x ) outside K, , that coincides with the harmonic function vp(x) inside the circle K , that assumes the same values as does u ( x ) on the boundary of K , , is an admissible comparison function [this being a consequence of the assumptions on the properties of the elements of the space W i ( Q ) ] .Therefore, f ( u ) 4 f (UP).
(7.3)
It follows from this inequality, inequality (7.2), and the definition of the function u p that
or, more briefly,
where p=.-f-. In Yp, we introduce polar coordinates ( r , 0) with origin at the center of K R . Let us expand u and wp in Fourier series with respect to 9: u (r, O) =
2
+
m
[a,( r ) cos ke
& =1
+ if)'
+b, ( r )sin k91.
m
el=-
WP(~,
[ a , (p) cos kO
&= 1
r
+ b, (p) sin he],
4 R.
458
OTHER D E V I C E S FOR OBTAINING BOUNDS
and that
From these equations and inequality (7.4), it follows that m
9 (P) ,< bk2 'k =l
[a: (P)4-bi (P)] ,< P P ~ (' P). p
,< R .
(7.5)
Let us integrate this inequality; more precisely, let us represent it in the form
and integrate from r to R:
This inequality yields the desired inequality
from which, on the basis of Lemma 4.1 of Chapter 2, we get the inequality
and the possibility of finding a bound for I U ~ ~ , ~where ,, Q'cQ and 1 a=--- 28 , in terms of [1u11,; (51), p, and the distance from Q' to the boundary S. This method of finding a bound for I U I ~ , ~ is , due to Morrey [36]. It is simple and esthetic, though applicable only to the case m = n = 2.
10 Other Boundarv-Value Problems 1. FORMULATION OF THE PROBLEMS AND THE GENERAL PROCEDURE FOR SOLVING THEM
In the preceding chapters, we considered primarily the first boundary-value problem. Only Sections 3 and 6 of Chapter 3 are devoted to the other boundary-value problems (and then for linear equations). However, most of the various a priori bounds given above for the solutions of linear and quasilinear equations are independent of the boundary conditions (these are the so-called interior bounds). Hence, they are equally applicable to all the boundary-value problems. Therefore, to investigate the other boundary-value problems, we need only study the behavior of their possible solutions in a neighborhood of the boundary. In the linear case and even in the nonlinear case in which the derivatives of the solution u appear linearly in the boundary condition, the entire investigation can be carried out basically along the same outline as for the f i r s t boundary-value problem. The r w d w will see below how this can be done, The situation is more complicated with the case of the general, nonlinear, boundary condition. To study it, we have needed to resort, not to the Leray-Schauder theorem but to another topological theorem regarding the existence of solutions of abstract equations of a definite class and to confine ourselves to the case in which the uniqueness theorem is applicable. Remembering that it is already time to bring our book to a close, we shall confine ourselves to an exposition of the most difficult and specific case in which the boundary condition depends on u, in a nonlinear manner. For the case of linear dependence of the boundary condition on u,, we give the necessary outlines that will enable the reader himself to c a r r y out the entire reasoning, using the a priori bounds given below.
459
4 60
OTHER BOUNDARY-VALUE
PROBLEMS
W e begin with a preliminary analysis of the boundary-value problems in which we are interested and with a description of the basic plan of solving them. In a bounded region 51, let us look at problems of the following form:*
Lu =a,/
LCs)u=[ b ( x , U. u,)+
bi ( x .
(1.1)
a ( x , u . u,) = 0,
( x . u , us) u,,,/+ U ) us,+
Bo(x,
U) 11s = 'p ( x ) l s
(1.2)
under the assumptions that Eq. (1.1) is uniformly elliptic and that, for arbitrarily fixed u and p , the factor I ( x , u, p ) with components b,, ( x , u , p ) + bi ( x , u ) does not, for any x S, lie in the plane tangent to S. More precisely, we shall assume that
e
[bPi(X,u ,
p)+b,(x*
U)]COS(n*
x,)>/v,(IuI.
IPl).
>o.
(1.3)
Just as in the case of the first boundary-value problem, we shall strive to reduce the question to the question of the solvability ofthe problems (1.1) and (1.2) to the problems of the existence of fixed points under certain transformations possessing a number of desirable properties. To do this, let us write the boundary condition (1.2) in the form
and let us consider the following problems:
where
ai/( x ) = a,/ ( x ,
'v ( x ) ,
w, ( x ) ).
a^ ( x ) = a ( x ,
w ( x ) , w, ( x ) ) ,
1
and v ( x ) is an arbitrary function in Cl,a(a). The problem (1.5) is linear in the unknown function u ( x ) . Let us suppose that the *This way of writing condition (1.2) is convenient for formulating the restrictions encountered below on the functions appearing in the boundary condition.
FORMULATION OF THE P R O B L E M S
461
uniqueness theorem holds for it. Then, u ( x ) is uniquely determined by v ( x ) . We represent this (in general) nonlinear correspondence (transformation) as follows: u = @ (v).
(106) The fixed points of this transformation are solutions of the problem (1.1) (1.2). Let us see what the properties of the transformation @ are. If V ( X ) E C ~ , . ( Q and ). i f all the known functions 82'8 %,(X, u , PI, a@, us PI, b ( x , u , PI, bi ( x , _ u ) , bo ( X , u)l and sufficiently smooth, then the functions a i l ( x ) and a ( x ) belong to C 0 , . @ ) , and, consequently, there are grounds for expecting the problem (1.5) to have a solution u ( x ) in C2,.@). However, in the general case, the functions g l ( x ) belong only to Co,a(since they are formed from the v X l ) . Therefore, we may not assert that a solution of the problem (1.5) will belong to the space C,,.(G) or even to C , , m + e ( G )where , E > 0 (cf. Section 3,Chapter 3). Therefore, we cannot show that the transformation CI, is completely continuous (that is, that v ( x ) is appreciably improved) and we may not apply to this transformation the Leray-Schauder criterion, for example, regarding the existence of fixed points. There is one case i n which this can be done, namely, the case in whichb(x, u . u,)=O. Then, the functions 4 ( x ) = bi ( x , v ( x ) ) and { ( x ) are elements of Cl, and the theorems in Section 3 of Chapter 3, which assert that the solution u ( x ) belongs to C2,=@), can be applied to the problem (1.5). We can reason for this case just as for the first boundary-value problem. The basic analytic part is the obtainingof a priori bounds on the solutions of this problem in the norm Cl,*(Q). We shall now obtain such bounds for solutions of the generalproblem (l.l),(1.2). To prove the existence of a solution of the abstract equation corresponding to the general case of problem (l.l),(1.2), we use the following theorem (cf. [31], Section 41). Theorem 1.1. Let C and E' denote two Banach spaces. Let
-
d=5
(a, 7)
denote a continuous transformation that assigns to elements a of C elements of of C' and that is continuously dependent on a real parameter T E [0, 11. Let us suppose that the folpwing conditions are satisfied for some connected closed set C, of the space C': (1) f o r every T in [O, I], the transformation 2 has locally a unique inverse in a neighborhood of an arbitrary pair of corresponding points: (2) a in 2 is compact i f , for all T E [O, I], it is mapped into a compact set in LA; (3) foy some particular value T = T ~ E [O, 11, at least one point in Cu is the image of only one point in C .
462
OTHER BOUNDARY-VALUE
PROBLEMS
Then, for every T E [0, 11, an arbitrary point in C i is the image of only one point in C. This assertion in the book by Miranda [31]was used to prove
the theorem on the existence and uniqueness of the solution to the Dirichlet problem for nonlinear elliptic equations. (We emphasize that, in contrast with the Leray-Schauder Theorem 1.1, it cannot be applied to problems with more than one solution.) Fiorenza [52] used it to study boundary-value problems of other forms. Specifically, using it to obtain bounds for the solutions of linear problems (cf. Section 3, Chapter 3), Fiorenza proved the following theorem on the solvability of the problem (l.l), (1.2): Theorem 1.2. Let us assume that the followingrequirements are
satisfied: (a) The coefficients where
nii(x, u. p )
and a ( x .
W : { x E M , lul,<M.
u,
p ) belong to Cl,,,(?lR),
I P I ~ M ~ } ,
and their partial derivatives with respect to u and the p k on satisfy a Lipschitz condition with respect to u and the p a . The functionb(x, u . p ) belongs to C 2 , m ( W and ) , its second partial derivatives with respect to u and the p k on W satisfy a Lipschitz condition with respect to N and the p k . The functions b i ( x . u ) and b, ( x . u ) belong to C2, (!Jn). The function 'Q ( x )6 Cl, (a). (b) S is a surface in the class C2,a. (c) Equation (1.1) is elliptic; that is, on W , DI
n
a , j ( x s u. P ) t l € j > / v ~ E : . v > o . i= I
(d) The functions b ( x , u , p ) and the I>, ( x , u ) satisfy condition (1.3). Suppose, in addition, that there are two families of differential operators L, and L!') of the same form as L and L(s)in(1.1) and (12) that depend continuously on the parameter T E [0, I] uniformly with respect to T in [0, I], that satisfy conditions (a) - (d), and that have the property that the problem
L,u = 0 , L?U
(1.7)
= [PIs
for
T = 1 coincides with the problem ( l . l ) ,(1.2) and, f o r has a unique solution u in C2,a (G) for s m e p ( x ) in Cl, (GI.
T
= 0,
Suppose also that, at an arbitrary solution u ( x , T) E C2, (a) of the problem (1.0,the corresponding problem obtained by unbounded variation of u is solvable and that, for an arbitrary family [pk ( x ) ] with uniformly bounded norms IpR11, 8 , the norms of the I
II
ol,
4 63
FORMULATION OF THE PROBLEMS
solutions Iuk ( x . T)[], Q, are Uniformly bounded with respect to k and T E [0, 11.* Suppose that
Then, for arfiitrary cp ( x ) in Cl,(a), the problem (1.7) has a unique solution u ( x , T) in Cz,o. (ST)f o r every T in [o, I]. To reduce this theorem to the preceding one, we introduce two Banach spaces: For C, we take Cz, (G)and, for E’, we take the space of pairs of elements a’ = [f,cp), where f E C o , a ( a ) ,cpE C , , a ( S ) , and LI
We define the transformation%(a,
a s f O l l O W 6 :To every a IJ ( x ) in C, we assign the element a’= (L,v; L%}. The set 2; is the set a’ of all elements in C‘of theform(0, ‘p (s)]. Obviously, by virtue of the assumptions made, the transformation 2 satisfies the requirements (1) and (3)of Theorem 1.1. Let u s show that it meets requirement (2). We shall do this for the value T = 1 since we are assuming that the differential operators L, and L?) possess uniformly (with respect to T in [0, 11) the same properties a s for T = 1. Thus, let us look a t some sequence { y k } that converges in C,, (S). Suppose that U ~ ( X ) E C ~is, ~a( solution ~ ) of the problem (l.l), (1.2) with v=cpk. From the conditions of the theorem, it follows that the norms l u k 11, a, are uniformly bounded. Consequently, there exists a subsequence (which we shall also denote by ( u b ( x ) J )that converges in the norm of C,(8). W e need to show that the u , ( x ) , for k = 1, 2, , form a convergent sequence in C ,=@). Consider the difference v = u - z of any two functions in the ( u k ( x ) ] . This difference satisfies the linear equation T)
LI
...
-
-
- -
-
Lu =Lu -Lu =a,,( x )u,,z, + a, ( x ) V X l+ a ( x ) 2, = 0 and the boundary condition
-
LCS’v 3 LCS’U
- LCS’u-
-
0i (x ) V X ,
tm
2, = cp
(1-8)
(4- a x ) *
where *More precisely, a s is shown in [52], uniform boundedness and equicontinuity of ( x , T) are sufficient.
uk ( x , r) and
4 64
OTHER BOUNDARY-VALUE
PROBLEMS
Here, d ( X ) =t u ( x ) +
(1
- t>u( X ) .
For definiteness, let u s suppose that 1 u 12,
> I -.:1
a, e . By using condition (a), one can easily show that, forthe coefficients of and
Z(S)
,
*
(alp
- -
b,. bla, y
","" Ilit a"l+ Igi.I ;
I&*
1, 0, e
4 c*
sc (Iu
12.0,
e
+
a u , s + l ~ lq* a , Y 4 C ( l 4 2 , . , P +
=,
1 4 2 , 0,s+
1).
1)
z
(1.9)
where the constant c depends on the norms . .1 iI1,a,e. (These last a r e , by assumption, uniformly bounded for the entire sequence ( u R ) . ) Keeping conditions @) (d) in mind also, we can apply to V ( X ) inequality (3.5) of Chapter 3 regarding linear equations. This, together with (1.9), yields
-
la 12, a, 9 sc I 'p -'911.0, s
+c (I
u 12, a, e
+ 1) m y I
+ Irp--rpll,~,,+~mpaxI~l[l~l,,a,e+l + 9F]+c l42,0,Y+
'p
-(PI +
+(lul,,o,,+
l"I,,o,P[1~1*,.,P+~+
+(I ulz,o, *+1 4 2 , O . * ) l + a l *
Then, on the basis of inequality (2.1) of Chapter 3, we have
465
QUASI LINEAR EQUATIONS
from which we get the inequality
IvI2, s,p Q c (1
+I u ~ ,
a,*)
[IvI 1,0, e+
IQ -?I,,
a,
s]*
(1.10)
From (1.10). we may conclude, first of all, that the norms k = 1, 2 , are uniformly bounded. This is true because, otherwise, there would be a subsequence ( U k i ] such that
...
I' k 12, a, Q' for
I u&i I,
a,
I ukf -
la
1 12. a , p'
and then, by setting V=uk
f
-111
1-1'
in ( l . l O ) , we would arrive at the inequalities
I u% I*,
a,
B
4
I vki 12,
<*'('+
a, p
Q
I'kt12,a,p)
[IwkiI1,o,P+I'~i-'k~-~
II,a,s]'
which are not valid since the expression in the square brackets approaches 0 a s hi+-. But it follows immediately from the and inequality (1.10) that boundedness of the norms I uk 12,., lU&-U112,a,D=
I~&,I12,a,9"o
as k, L-+c%
which completes the proof of Theorem 1.2. This theorem 'reduces the investigation of the solvability of the boundary-value problems (l.l), (1.2) to the search for a priori bounds of the solutions in the norm of C l , a ( Q ) .In the following sections, we shall describe those cases in which mathematicians have succeeded in finding such bounds (cf. [62]).
2. QUASILINEAR EQUATIONS WITH PRINCIPAL PART IN DIVERGENCE FORM Consider the following quasilinear equation with principal part in divergence form L,(u)= - (da , dXi
( x , u . u x ) ) + a ( x , u , u.) = 0.
(2.1)
Associated with this equation is a boundary condition of the form [ a i ( x , u , u,)cos(n, x i ) + y ( x ,
Let us assume that the function function Q ( x , u ) defined for all x E G.
U)](~=O.
cp(x. u ) I s
(2.2)
is the value of some
466
OTHER BOUNDARY-VALUE
PROB LEM S
It will be interesting to prove an existence theorem for the problem ( 2 3 , (2.2), with natural restrictions on the a , and u of the same type as in the case of the Dirichlet problem (cf. Chapter 4). A s shown above, to do this, we need to obtain a firion. bounds on the solutions in the norm of Cl, o. (a) under natural restrictions on the 0 , and a . For an arbitrary interior subregion 51' c Q , such bounds have already been obtained (cf. Sections 1-6, Chapter 4). Therefore, we shall now seek bounds close to the boundary S, assuming that condition (2.2) is satisfied on S. For the moment, we shall not go into the question of a bound for the maximum absolute value of the solution u ( x ) itself but shall find bounds successively for I u J ( ~ ) ,e , max JVul,and Iuxk I(=), *, 8 assuming that max lu(x)l<M and u ( x ) E C 2 ( G ) . e
Let us assume that the functions a,(%, u . p ) and a(%. u. p ) , for x E n, 1u I hl, and arbitrary p , are differentiable with respect to their arguments and satisfy the natural restrictions
We assume that the boundary S ofthe region 8 is twice continuously differentiable and that the function ~ ( x u. ) belongs to
cl,dlx€fL lul,<Ml>. Let S, denote an arbitrary fixed portion of the boundary S and let us find the desired bounds in a small region 51, adjacent to S,. A s a preliminary, let us "flatten out" S , by making the change of , n. Suppose that the independent variables y, = y i ( x j for 1 = 1, equation for S, in the new coordinates is y,, = 0. Equation (2.1) and condition (2.2) take the forms
...
(2.1') (2.2 ')
where
467
QUASI LIN EAR EQUATIONS
W e conclude from (2.1') and (2.2') thatu (x)satisfies the integral identity
for arbitrary q in W:(8). Suppose that q vanishes on S \ S,. Then, if we transform the surface integral in (2.5) to a volume integral, we obtain the integral identity
-
In (2.6), let u s set q = C" max ( u k ,0),where k is an'arbitrary number and C (y) is a smooth function of compact support with range in [ 0, 11 on an arbitrary sphere K , thatdoes not intersect S \ S, and that has its center on S,. Thus, on the basis of (2.3) and (2.4), for k >/ m a x u -6 ,we obtain the following inequality just as in Section 1 K,nQ
of Chapter 4.
I v u I"C*dY ,< 7 'k, p
[
(u-k)"I
vcImdY f mes Ak,
,].
(2.7)
'k, p
Here, Ak, is the subset of K n 8 on which u (y) > k and 6 > 0 is some number determined by A, the constants in inequalities (2.3) and (2.4), and the conditions on 'p and S . Analogously, by setting q = C" max 1- u - k . 0) in (2.6) ,we arrive at inequalities (2.7) for the function - u ( y ) when k
>rnax (-
u ) - 8.
KpnP
W e note that inequalities (2.7) for u and -u were proven in the case of the first boundary-value problems under the supplementary assumptions
respectively. Here, we do not need these restrictions. From (2.7), we derive a bound for the norm IuIa close to S1 on the basis of Theorem 7.2 of Chapter 2. Let us suppose that we have found a bound for maxIVuI, and let Q
u s find a bound for I u, I where 51, c 9: u S1. In the-identity (2.6) , 4 (Q), 98' we set
4 68
OTHER BOUNDARY-VALUE
PROBLEMS
where E is an arbitrary sufficiently smooth function that vanishes close to S\S,, and we transform the result as follows:
Obviously, this identity will remain valid for any E ( y ) in W: ( Q ) that vanishes on S\ S,. Denote by Ak,,the sets corresponding to the function uyr for some fixed r f n . In (2.8), we take first E=C2max(uy - k , 01, t h e n € = C2 max {- uyr- k . 0) with arbitrary k , in view of the boundedness of maxIVu) and condition (2.3), we derive the inequalities r = l , ..., n - I , and analogous inequalities for - u y r . These inequalities a r e valid for arbitrary k . From this and from Theorem 7.2 of Chapter 2, we obtain a bound for Iuyrla,&,, for r # n. W e then get a bound in C , . ( Q , ) for the derivative along the normal to S, with the aid of Eq. (2.1’) and inequalities (2.9). Specifically, it follows from (2.9) with k = min uyr that Kpng
J I vuYr 12 d y ,< r p n - 2 + 2 a S
r
+n (2.10)
Kpn*
for spheres Up having no points incommonwith S\S,. hand, when we solve Eq. (2.1’) for uYny,we see that
On the other
with bounded b,, and b. From this, it is clear that inequalities (2.10) are also valid when r = n. Therefore, on the basis of Lemma 4.2 of Chapter 2, the norm [ uy, I., g , , where Q , c 51 u S,, is bounded in terms of the constants ‘I and a in (2.10) and the distance from 51, to S\S,. Let us see about a bound for max lVul in the portion 8, of the region Q belonging to S,. Just as in the case of the first boundaryvalue problem, to get a bound for rnax IVul, we first get a bound for Qi
(2.11)
469
QUASI LINEAR EQUATIONS
with arbitrary 1 > 0 and then a bound for the maximum absolute value of the gradient of u ( x ) on S. In the case of the first boundaryvalue problem, this last bound was obtained byBernstein's method, which rested on the fact that we know the first derivatives of u in the tangential directions on S from the first boundary condition. In the present case, i f we had succeeded in finding a maximum on S of the derivatives of u in the tangential directions, we would have immediately been able to exhibit a bound for the derivative along the normal, and hence for max I V u ( , by using the boundary S1 condition. Specifically, (2.1') yields
W i t h the aid of conditions (2.3) and (2.4), we easily derive from
*I
in terms of max lul and max max 1 u,, I. r+n S1 Thus, we need to find bounds for the integrals (2.11) and the maximum absolute value of the derivatives u,,, for r # n, on S,. W e begin by finding bounds for the integrals (2.11). First of all, by considering (2.6) with q = e W n , where C is a smooth function of compact support with range in [0, 11 on some sphere K, that does not intersect S\S, and A is a large parameter, we see that (2.12) a bound for
d y n sI
s1
J (VulrndyS.(v,
p* m)
(2.13)
81
with v = v ( M ) , H = p ( M ) , and rn in (2.3) and (2.4). To obtain bounds for (2.11) with 1 > rn, we need to use, in addition to Eq. (2.6), the identity (2.8), in which we integrate by parts:
r=l.
(2.14)
..., a-1.
Here, the quantity
& = -( a i y k x f Y k
+ ': + $Yn
does not increase faster than the IVUl.
ai
d2Y,
dxldy,
(rn - 1) st
power of I Vu I for large
4 70
OTHER BOUNDARY-VALUE
PROBLEMS
W e define "-1
n
In the identity (2.6), we set 7 = [ u (y)-
u (yo)]wS+'w~C2, s. q
>0.
where C ( y ) is a function as above on the sphere Upwith center yo ES, and with radius p not exceeding the distance between yo and S\S,. The identity (2.6) then takes the form
J {aiUXIws+lw'C2 +(U - u (yo) ) [ a t (S + 1) YsoXi W C ~+
Q
P
+alws+'qwq-'w aiws'-'wq2CC +bwS+'wqc2+ +$ (s + 1) W'wy,w~C?++ w ~ + ~ q w ~ -,% J+*~+~w92CCy,J + xi
c2+
x1
y c2
+ + u ~ , w ~ + ~ ~ ~d yC = ~ )0,
(2.15)
where b=a
Iaxi dly dy, a+$uUY,+$Y,.
It follows from our assumptions that IbW
P)l
IPIY-
(2.16)
Noting that, on the basis of (2.3), a , @ , us P ) P f
> vo(l~l)IPlrn-P0(l~I)* v o > o
(2.17)
we use conditions (2.3), (2.4), and (2.16) to find bounds for the quantities in Eq. (2.15). We obtain the inequality rn
471
QUASI LINEAR EQUATIONS
where cl is determined by the constants 1 ( M ) , (2.4), (2.16), and (2.17), by the quantity max
~ € 9 1111 , ~< M
I
Jcu* $Y*
,and
po(M)
v,(M)
in
I
and by the boundary S. Let us apply Cauchy's inequality (1.2) of Chapter 2 with E = 1 to the t e r m s on the right side of the above inequality and let u s use the fact that max I - u (Yo) I 4 %Pa. P
Then, for p < (C,C,)-~,
1
we have
where c is a known constant. Let u s not consider Eq. (2.14), taking E equal to uyrwswQC2, where r f n :
Summing from r = 1 to n positive t e r m s
- 1, we get expressions for the two
W e use inequality (2.3) to bound these terms. W e transpose the other t e r m s to the right. We find bounds for their absolute values, noting that, by virtue of condition (2.4),
472
OTHER BOUNDARY-VALUE
PROBLEMS
This yields
If we now apply inequality (1.2) of Chapter 2, we obtain, for
q = s = 0, the inequality
and, for arbitrary s > 0 and q
r
> 0, the inequalities
rn
+(I +s)d+'wg+.e]
dy.
From inequality (2.18) with q = s = 0 and inequality (2.19), it follows that, for p not exceeding some number po determined by a and the constants in (2.18) and (2.19),
QUASI LINEAR EQUATIONS
rn
4 73
m -
Since ow?-' 4 w 2 , we conclude from the above inequality .on the basis of inequality (2.13) that (2.21)
Suppose now that s > 0. It is easy to see that (2.22)
since
Therefore,
and the derivative uynyn can be expressed from the equation in t e r m s of the other derivatives. We see, on the basis of (2.3) and (2.4), that (2.23)
By using (2.22), we easily conclude from (2.18) and (2.20) that, for sufficiently small p (less than some po > 0) and for q such that the ratio q 2 / s is much less than unity,
where the constants c ( s . q) and po depend on s and q. Consider the sequence of regions Qp,, where p, = (for s = 1, 2 , .) and let C, denote a smooth function of compact
..
%+&,
4 74
OTHER BOUNDARY-VALUE
PROBLEMS
support with range [0, 11 on the sphere Kps that is equal to unity in KP,+,. Taking q equal to 0, weconclude from (2.24) and (2.21) that
J u S + ' w ~ Cdy: < c (s) 11'
%
for arbitrary s > 0. Then, by induction on q = 0, 1, the integrals
j
(
ys+,wq+;
+'
m-2
2
L I ~ w ~ u;.~) + ~ dy
QPP
...,we see that
,< c (s,
q),
(2.25)
rffl
are bounded. Here, the constant c(s, q)+m as s, q+m. Taking (2.23) into consideration, we finally may write (2.26)
Let us now see about a bound for the tangential derivatives close to S,. W e first findaboundfor the maximum of the function z='uc2, where C(y) is a function for the sphere KPpas above. In the case m >/ 2, we may use the same procedure as we used in Section 3 of Chapter 4 to find a bound for lVul within P. Specifically, in Eq. (2.8), we set E=uyrC2max ( z ( y ) - k ,
0).
rfn,
k>O,
and we denote by A, the set ofpoints y E 8 for which z ( y ) > k . This yields
Since
it follows from (2.27) on the basis of conditions (2.3) and (2.4) that, for rn> 2,
QUASI LINEAR EOUATIONS
4 75
The constant c, depends on v ( M ) and p ( M ) in conditions (2.3) and (2.4) and also on the radius p/2 of the sphere KpD.However, this
last is of no significance in the present case since the sphere Kpp is fixed. Let us apply Hijlder's inequality with exponents p=& and p'=where 1 > to the integral I-m-4'
vn
Then, on the basis of (2.26) proven above, we arrive at the inequalities
From these inequalites, we get, on the basis of Lemma 5.3 of Chapter 2, a bound for max z = max vC2 P
P
with m>, 2. In the case 1 < m < 2, in order to get a bound for m m z , we set, in (2.8), E = uy,vswC2rnax [ z (y) - k, 0).
The basic t e r m s in the resulting integrand are
4 76
OTHER BOUNDARY-VALUE
PROBLEMS
W e find a bound for the term f3, which contains the t e r m wxl,by using Cauchy's inequalities (l.l), (1.2) of Chapter 2 and inequality (2.22) of the present chapter:
4 y1 fz+ Here,
V=
v (M)
and
gf l +
$( 1 + IVul)m-2w~vs+l(z- k)C? a r e the constants in conditions (2.3)
p = p(M)
and (2.4). If we take s sufficiently large so that fairly easily derive the inequalities s ( 1 + ~ V U ( ) ~ - ~ Vz12dy V~W(
< c1
(1
< 1, we can
+ 1Vu()m+6+2sdy,
Ah
and hence the inequalites
sI
Vz l2 d y
< c1
Ak
(1
+ I Vu
2s dY.
Ak
On the basis of (2.26) with 1 > above, ensure that
mf6+2s
inax z = max P
Thus, we have shown that I
~
n, these inequalities, just a s
(r 2 uf + 1) =1 r n-I
t2.
I uyr X I < MI
for r # n. In so doing we have, as indicated above, found a bound for max I Vu I. In addition, inequalities (2.26) a r e valid. All this S!
enables us to find a bound for max I Vu I close to S1, as was done in the first boundary-value problem (see Section 4 of Chapter 4). Let us formulate the assertions that we have just proven regarding bounds for u ( x ) as Theorem 2.1. Suppose that the following conditions are satisfied: (a) T h e j k c t i o n s a , ( x , u , p ) and a ( x , u , p ) , f o r X E Q , 1u1 Q M , and arbitrary p , are differentiable with respect to their arguments and they satisfy inequalities (2.3) and (2.4). (b) u > E c l , l I ~ E Q Il u I 4 M l and cP(x1
x E Q.max IUI< M
I w vxtUlcPxlxjl
.P,',
c~~~~
(9.1
(c) The boundary S of the region P belongs to C,.
(2.28)
4 77
QUASI LINEAR EQUATIONS
Let u ( x ) denote a twice continuously differentiable solution of the problem (2.1), (2.2) such that maxl u I ,< M
Then, the n o m I u II,.,& is, for some positive a , bounded by a constant depending only on M , on the constants m , v ( M ) , and p(M)in (2.3), (2.4), and (2.28), and on the boundary S. The subscriPt a i s determined by the same quantities. These inequalities are sufficient for us to be able, on the basis of Theorem 1.2, to assert the following theorem on the solvability of the problem (2.1), (2.2): Theorem 2.2. The problem (2.1), (2.2) has a unique solution u ( x )
belonging to Czs (a) i f the following conditions are satisfied: (a) For all possible solutions u ( x , T) E C , (a)of the boundaryvalue problems I
D.
(2.2 9)
where
the quantity max 1 u ( n , ~ ) l ,f o r T E [O, 11, does not exceed some 9 number M. (b) For x E G, I u I ,< M , and arbitrary p , the functions a, ( x , u , p ) and a ( x . U , p ) are diflerentiable and they satisfy conditions (2.3) and (2.4);y ( x , u ) satisfies (2.28). (c) Let us denote by %, the set ( X E a
IuI<MM.
IPI
<MIIS
where M , is the least upper boundof absolute values of the gradients ) the problems (2.29). of all possible solutions u ( x , T ) E C ~ , , ( G of (Note that, in accordance with Theorem 2.1, the quantity M , is
determined only by M , by the constants nz, v ( M ) , and p ( M ) in (2.3), (2.4), and (2.28), and by the boundary S.) The functions u i ( x , u , p ) must belong to CZ,II ( W * a (x.
f-t,
P) E CI, a
Also the first partial derivatives of
a
‘p (XI u )
E cz,
(L
(?m.
4 78
O T H E R BOUNDARY-VALUE P R O 6 L E M S
and a with respect to u and the p,: must satisfy a Lipschitz condition on m with respect to u and the .uk. (d) The boundary S is a surface in the class C2sm. (e) For an arbitrary solution of the problem (2.29) belonging to C2, (n), the corresponding variational problem is solvable. Remark: It is easy to show that, if y ( x , u ) i s of the form b, ( x , u) + 9 ( x ) , to prove the conclusion of the theorem, we need only assume that
4 ( x , 4 E c 2 , o (W*and
$ ( x ) E c,, a
(a).
Clarification of the conditions under which requirements (a)and (e) are satisfied is made in the general case by studying the spectrum of the differential operator L in (2.1) defined on the s e t of functions satisfying condition (2.2). We can give sufficient conditions for (a) and (e) to be satisfied. For example, we have the following lemmas: l e m m a 2.1. Suppose that the inequalities dai
dpl
p , El€,
>0
for x
E S, 1pI < 00
and x
E Q,
(2.30)
p = 0,
(2.31) u [ a ,( x , u , O)cos(n, x,)+y(x,
(2.32)
u ) l > 0, x ES
>
are satisfied f o r sufficiently large I u I R. Then, for an arbitrary solution u ( x , T) of the problem (2.29) in C, (Q)n C,
(a),
W e suppose also that S belongs to 0,, that the derivatives
exast and that they are bounded f o r finite values of u. The proof is simple. It rests on the principle of the maximum absolute value for elliptic equations. First, we use (2.31) and (2.30) to show that the solution cannot have a maximum greater than M or a minimum less that - M within Q . Then we find a bound for max I u I . S
Let us clarify this last step. W e note that, for T = 0, the problem has the unique solution u E 0. Suppose that, for some T > 0 the maximum of the function u ( x , T) is greater than R and that this maximum is attained at a point A, on the boundary S. By intro, y,, let us transform a portion S, of the ducing coordinates y,, boundary in a neighborhood of A,, in the plane y, = 0 in such a way
...
W A S I L I N E A R EQUATl O N S
4 79
that the outer normal at the point A, coincides with the y,-axis. Then, the boundary condition in (2.29) takes the form
where
A t the point A,, the derivatives of the form uyr, where r f n , vanish, and uy,>/ 0. Therefore, on the basis of condition (2.30), we have at the point A,
which, for u > R , contradicts (2.23). Analogous reasoning at a point a t which u ( x , T) has a minimum leads to a bound for u from below. l e m m a 2.2. Suppose that the a i ( x . U , p ) and a ( x , U , p ) satisfy the conditions of Theorem 7.1 of Chapter 4 and that, for X C S and sufficiently large Iu I >/ R , "'p
(x, u )
> 0.
(2.33)
Then, f o r any solution u ( x , r) in Wk (Q) n L (Q), where n > m > 1 q > m n ( n - m ) , of the problem (2.29), !he quantity essential max I u J is bounded by a constant depending on R , on mes Q, on e ( ( U I I ~ ~ ( ~ ) . on the constants v,, E , and a i r on ~ l ' p f ~ ~ L (where r j ~ e ~ 1 = 1 , 2, 3), on the conditions of Theorem 7.1 of Chapter 4, and on the boundary S , which is assumed to be piecewise-smooth.
and
The proof of this lemma is analogous to the proof of Theorem 7.1 of Chapter 4. Specifically, when k > H, inequalities (7.8) of Chapter 4 are valid for u ( x , r) and -u ( x , 5 ) . From these inequalities we get a bound for essential max IuI on the basis of Theorem 5.2 e
of Chapter 2. These inequalities a r e derived from the integral identity (corresponding to (2.29)
480
OTHER BOUNDARY-VALUE
PROBLEMS
+ 7 ( x i E wf, (Q)n L, (51).
T
'f7 ds = 0,
S
-
in which we take 7 equal to max ( u -k, 0) or max ( - u k, 0). Here, on the basis of (2.33), the boundary integral has, for k > R , the sign necessary for the bounding inequalities. l e m m a 2.3. Condition (e) of Theorem 2.2 will be satisfied if (a) Inequalities rnax l u l < Mand max )Vu(<M,,whereM , and M, e P aye positive constants, are valid f o r all solutions u ( x . T) in C,,,@) of the problem (2.29), where T E [O, 11. (b) The functions a,, a, and 'p are the same a s in hypothesis (c) of Theorem 2.2. (c) The inequalities (2.34)
where
are valid in the region
(d) SEC,,.. Proof: Let us consider the variational problem corresponding to
(2.2 9) :
48 1
QUASI LINEAR SYSTEMS
A t an arbitrary solution u ( x , T)EC,.(G) of the problem (2.29), this will be a linear problem in 6u of the form (3.8) of Chapter 3. Also, the coefficients in the equation and the boundary condition will, by virtue of (b), belong to C o , a ( a )and Cl, = ( S )respectively. Therefore, on the basis of Theorem 3.2 of Chapter 3, we may assert that the problem (2.36) has a unique solution in C2, (g)for and 9 E Cl, (S) if and only if the corresponding arbitrary f E Co,
(a)
L1
homogenous problem has only the trivial solution. Therefore, let us turn to the integral identity
corresponding to the homogeneous problem (2.36). If we set q = 8u in this identity and use condition (2.34) and Cauchy's inequality with c = v /2, we arrive at the inequality
From this, we conclude on the basis of condition (c) that 8u = 0. Therefore, the inhomogeneous problem (2.36) does indeed have a unique solution for arbitrary f and
+.
3. QUASILINEAR SYSTEMS Let us look at elliptic systems of the form L U E L Z , ~ ( X ,~ ) u x , x , + b i ( x , U, u , ) u ~ , + b ( x ,
U.
u,)=O.
-
(3.1)
where the a,,, the b,, and b satisfy inequalities (2.1) (2.4) of Chapter 8. Let us take a boundary condition of the form
482
OTHER BOUNDARY-VALUE
PROBLEMS
A Priori bounds on the norms I U I ~ , ~ of , ~ solutions u ( x ) of the problem (3.1), (3.2) have also been established in t e r m s of max Iu(
(cf. [621).
For arbitrary Q ' c Q the , bounds on I U ~ , , ~ , W were discussed in detail in Chapter 8. In considering regions belonging to some portion S, of the boundary S, we need to modify in Chapter 8 in a manner corresponding to what was done in the preceding section for the case of a single equation. In particular, before finding a bound f o r the maximum value of w r l V u 1 2 1 close to S,, we need to find a bound f o r the maximum value on S, of the tangential gradient
+
fl-1
N
and to do this, we first get an inequality analogous to (2.26). A f t e r we have established that ~ I I ~ ~ . . , does Q not exceed some constant M,, we immediately get a bound for the norm Iu12,.,e from Theorem 3.1 of Chapter 3 since each of the components uf ( x ) of the vector-valued function u(x) can be regarded a s the solution of a linear problem of the form (3.1) of Chapter 3. Here, the norms in Co,a(G) of the coefficients in Eq. (3.1) and the norms in C,,,,(S) of the functions aI,, b, and '9 in condition (3.1) of Chapter 3 are bounded by a constant depending on M,. With the aid of inequalities of this type and the Leray-Schauder principle, we can prove Theorem 3.1. Suppose that, f o r all solutions u ( x ,T) E C2,&) of
the family of problems
L,u
+(1 -
TLU
T) (Au
-U) = 0,
I
the a priori inequality (3.4)
holds uniformly for T E [O, I] and that the following conditions are satisfied: (a) S E c2,a. (b) Thefwctioncp(x. u)definedontheset(xES, lul< MJbelongs to CIS1'
(c) Inequalities (2.1) - (2.4) of Chapter 8 are valid f o r the a i j , the b; the quantity E ( M ) i s determined by inequality (2.8) of Chapter 8. (d) The functions b, ( x , u, p) and b ( x , u, p ) defined on the set b,
, and
QUASILINEAR SYSTEMS
483
belong to C?,"(111), where M , i s a constant depending on v ( M ) , c1 ( M ) , and E ( M ) in conditions (2.1) - (2.4) and (2.8) of Chapter 8 such that maxlVu(x, e
T)I<
M,.
T E l O , 11.
Then, f o r every 7 in (0, l ) , there exists in C2,m(g) a solution T) of the problem (3.3).
u (x,
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Index
Dirichlet problem, 9, 14, 128,140,147,160,
Aleksandrov, A. D., 19, 30 Banach space, 3-5, 40, 41, 293, 461, 463 Barrier functions, 375 Bernstein, S. N., 29, 32-34, 191, 224, 225, 227, 266
Bernstein’s assumption, 191 device, 227 method, 266,469 Bers, L., 29, 32, 229 Birman, M. Sh. 185 Boundary integral, transformations of, 228 Boundary-value problems, 386 solvability of, 244 Bounded solutions, theory of, 246 Browder, F. E., 38,401
Caccioppoli, R., 28 Cartesian coordinate system, 6, 174 Cartesian product, 293 Cauchy’s inequality, 39, 40. 64- 143. 215. 218, 264, 265. 272 2i 476,481
Conditional variational problem, 321 solution of, 326 Continuation along a parameter, method of, 403
Conies, H. 0..30 d e Giorgi, E., 29, 55 d e Giorgi’s method, 446 result, 35 theorem, 28 Device of transformation, 224 Differentiability properties, 335,404 Diffraction problems, steady-state, 205 Direct methods, 320
183, 196, 209, 299, 309, 314,317, 370, 378. 402. 462. 466 classical iolv&ility of, 106, 107 Schauder’s theorem on, 106 solvability of, 236, 246, 291, 292, 317 uniqueness theorem for, 10,387 Douglis, A., 401
Egoroff’s theorem, 323 Electrostatic potentials, 113 Elliptic equations, behavior of solutions of, 93
differentiability properties of solution of, 209
linear, 244 quasilinear, 244 solutions of, 29 solutions of, 424 Elliptic operators, fundamental inequality for, 162, 169 Embedding theorems, 16, 249 Energy inequality for elliptic operators, 144 Euclidean space, 41, 239 n-dimensional, 43, 117 Euler equation, ellipticity of, 32, 332 Euler’s equations, 31, 35, 141, 244 f o r minimal surfaces, 314 generalized solution of, 326, 331, 335, 336
Existence theorems, 291, 297
Finite differences, method of, 65, 222 Fiorenza, R , 135,462 Fiorenza’s results, 28, 106 Fourier series, 457 Fredholm equations, second k i d , 223 Fredholm solvability, 208 493
4 94
INDEX
Fredholm’s alternative, 138 theorems, 14, 109, 116, 154, 156, 157, 160,388,403 Friedrichs, K. O., 29 Galerkin’s procedure, 182 Generalized solutions, proof of Holderness of, 331 theory of, 412 Guseva, 0. V., 185, 401 Gyunter, N. M., 113 Hamilton’s principle, 140, 141 Harnack’s inequality, 446 Heinz, E., 406 Hilbert problems, 337 space, 3, 31, 40, 154 Holder classes, 196,402 condition, 4, 37, 81, 82, 90, 91, 94, 102, 221, 231, 246, 373, 380 constant, 4, 17, 82, 90, 424 bounds for, 228,423 evaluation of, 131 inequality, 16, 40,44,46, 56, 62,75, 87, 143, 145, 165, 198, 200, 230, 288, 289, 329, 391, 400 norms, 68. 117, 310, 395, 397, bounds for, 228 svace. 383 Holhernkss of functions, 37, 100 Hopf, E., 29, 32, 34
100,
84, 265, 406
Injection theorems, 237 Jacobians of transformation, 6 John-Nirenberg lemma, 427 theorem, 446 Kondrashov, V. I., 43 Kronecker delta, 2, 204 Ladyzhenskaya, 0. A., 29 Laplace’s equation, 119, 129 Laplacian operator, 170, 180, 184, 226 Leray, J., 32, 385 Leray-Schauder criterion, 292, 461 theorem, 293, 298, 299, 384, 422, 459, 462,482 Linear equations, 106 solution of, 9 theorem on, 279 Linear systems, elliptic, 386 Lipschitz condition, 243, 305,462, 478 Majorization, method of, 375 Maximum absolute value, finding a bound for, 327
Maximum principle, vector analog of, 386 Mikhlin, S, G., 29 Miranda, C., 32, 135, 462 Miranda’s inequalities, 135 Miranda’s results, 28 Morrey, C. B., 28, 29, 34, 35, 36, 56, 425, 458 Morrey’s method, 425, 456 Moser, J.. 424 Moser’s method, 425,446,451
Nash, J., 29, 35 Nash’s method, 446 Natural restrictions, definition of, 336 Newtonian potential, 61, 117, 126, 128, 129 Nirenberg, L., 29, 32, 228, 229,385,401, 452 Nirenberg’s estimate, 452 method, 425
Parabolic equations, solutions of, 29 Petrovskiy, I. G., 191 Plotnikov, V. I., 32, 34 Poisson’s equation, 117, 140, 141 Quasilinear equations, solutions of, 9 Regular boundary conditions, 228 Riesz theorem on linearfunctionals, 29,152 Scalar functions, 387 Schauder, J., 28, 32, 34, 109, 110, 385 Schauder-Leray theorem, 378 Schauder’s a priori estimate, 116 Schauder’s “gluing” method, 30, 235 Schauder’s inequality, 184, 310 Schauder’s results, proof of, 106 Schauder’s theorem, 106, 107, 294,295,384 Sigalov, A. G., 28, 32, 34, 35 Skvortsov, G. E., 185 Slobodetskiy, L N.. 401 Sobolev, S. L., 43 Solomyak, T. B., 38 Solonnikov, V. A., 117 Stampacchia, G.. 29 Strengthened regularity condition, 321 Subsolutions, 202 Surfaces, saddle-shaped, 240, 243 properties of, 240 Symmetric operators, theory of extension of, 185 Tangential derivatives, boundedness of, 220 Holderness of, 220 Three-point condition, 239-243, 314 Two-dimensional problems, 379
495
INDEX
Uniform ellipticity, condition of, 379 Uniqueness in the small, theorem on, 60,
Vekua, I. A., 29, 229 Vishik, M. I., 29, 38 von Neumann, J., 243
Variational problems, 318 Vector-valued functions, 386, 387
Young’s inequality, 40, 46, 47, 248, 250,
225, 321, 334
287, 431, 432, 435, 437, 438
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