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162
163
Definition 8. An operator P is called regularly hyperbolic in X if for any point
R E X we can find a neighbourhood K and a number f,with the property that f < Ro if R E 8 u S , and f = Ro if R E $, such that the Cauchy problem (1.1)
+
(1.2) for the operator P + Q, where Q is any differential operator of order not exceeding m  1 with infinitely smooth coefficients, is C"well posed in K ; for T2f If the wellposedness index at every point R can be chosen independently of Q, then P is said to be completely regularly hyperbolic in X .
m
It is a simple matter to give a description of completely regularly hyperbolic operators. Theorem 12. I n order that P, an xohyperbolic operator in X , be completely regularly hyperbolic in X , it is necessary and suficient that d*,
(1.12)
for (x, 5 ) E X x (IR"+'\O). Moreover, the Cauchy problem for P is C"well posed in any spatial type lens 2 C X , and the following estimates hold. (i) If ?._ = T, T+ < T+ and x E [l, 2],12 s, then IIU; zrIIl+ml,s+m1
< C I I UXTIIl+m,s+ml ; (1.13)
uniformly in t, TE [T, T+],T < t. (ii) If ?r_ > T, ?+ = T+ and x E [2, a],1 2 s, then
!
(a) Q = &o(x:
c
(
IIU; z T l l l + m  l , s + m  l +
1;
IIPU; Xt'Ill,s
+ 5:), o > 0;
Q = C L X , ~ ~CL, > 0; (4 Q = 5:; (4 Q = 5:; (4 Q = 0; (f) Q = 5: + 25152 + d. (b)
dt')
(1.14)
uniformly in t, T E [f,?+I, T < t ; i f x = a, the lefthand side of (1.14) has to be replaced by IIU; z t I I l + m  l . s + m  1 ( T +
+
where prr, pxe; = 'psx and p,, are (n 1) x (n 1) matrices of secondorder derivatives, and I(.) denotes the transpose matrix. The fundamental matrix has the following properties (Ivrii and Petkov [1974], Hormander [1977]): 4 H i p , 4 ) = Fp(P)Hq(P)9 where Hq = (aq/ag)(a/ax) (aq/ax)(a/at)is the Hamiltonian field generated by q and { p, q } = Hpq are the Poisson brackets. b) The fundamental matrix is transformed by a canonical diffeomorphism into a similar matrix; thus its eigenvalues are symplectic invariants. c) The matrices Fp and  Fp are similar. d) If the symbol p is hyperbolic in X and p = (X, 4) is its critical point, then all the eigenvalues of the matrix FJp) are either pure imaginary or there are two nonzero simple real eigenvalues p(p) and  p ( p ) and the remaining eigenvalues are pure imginary. e) Moreover, if f o is a root, with multiplicity r, of the polynomial p(E, to,f'), then r 2 2. If X E 8 and r 2 3 or if E E S+ u S and r 2 4, then Fp(p)= 0. If E E S+ u S and r = 3, then either Fp(p)= 0 or rank Fp(j7) = 2 and Fp(p)has two nonzero real eigenvalues. f) Let Q(z) = $o(Fp(p)z,z), the localization of the symbol p at the point p, be a quadratic form in z E V = IR"" x IR"", where o is a bilinear symplectic form, o((x, <), ( y , q)) = (x, q)  ( y , 5). If r = 2, X E X , p is xohyperbolic in X and ptoc0(p)< 0, then V can be decomposed into a direct sum of symplectic orthogonal subspaces that are invariant under Fp(j7)and on each of which Q has one of the following canonical forms:
 t)1/2*
The necessity of condition (1.12) has been established by Ivrii and Petkov [1974] and its sufficiency by Ivrii [1976a] by the method of alternating operator. Corollary. If P is completely regularly hyperbolic in X , then it is strictly hyperbolic in X and the multiplicity of its characteristic roots does not exceed 2 if (x, 5 ' ) E (S+ u s) x (IR"\O).
Moreover, there is exactly one subspace of the type (b) u (d) u (f); the subspaces of the types (a)  (e) are two dimensional, while those of the type (f) four dimensional. To the subspace of the type (a) there correspond eigenvalues k i o of the fundamental matrix and to the type (b) the eigenvalues fp; the eigenvalue zero corresponds to the remaining types of subspaces.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
f') If, however, r = 3, X E S+ u S and p is x,hyperbolic in X , then in the decomposition of V there may be subspaces of types (b) and (e) only; moreover, the number of subspace of type (b) cannot exceed 1.
X E X and K be a sufficiently small neighbourhood of X that depends only on p. Zf
164
Definition 9. An x,hyperbolic operator P with real principal symbol p is called effectively hyperbolic if at any critical point p its fundamental matrix has nonzero real eigenvalues fp, p > 0. At critical points p, we also introduce the subprincipal symbol of the operator by the formula
. . .." 1
P"
= pm1
+ 2 1 pxjtj; j=o
this is also invariant. Ivrii and Petkov [19741 conjectured that an operator is regularly hyperbolic if and only if it is effectively hyperbolic, and established its validity in one direction.
Theorem 13. Zf the operator P is regularly hyperbolic, it is effectively hyperbolic. Corollary. Let the operator P be regularly hyperbolic in X . The multiplicity of its characteristic roots does not exceed 2 if (x, 5') E k x (lR"\O)and if (x, 5 ' ) E (S+u S) x (lR"\O), the multiplicity does not exceed 3. Theorem 14. Let P be effectively hyperbolic in X . Suppose that one of the following three additional conditions is satisfied: (i) x = 8,S+ = S = 0. (ii) p can be factorized in the neighbourhood of each of its critical points p, that is, p = p l p 2 in the neighbourhood of p, where p l , p 2 E C" and p l ( p ) = p 2 ( p ) = 0. (iii) p can be factorized in the neighbourhood of each of its critical points p = (x, 5 ) such that 5, is a characteristic root of multiplicity 3. Then P is regularly hyperbolic in X and the Cauchy problem is C"well posed in any spatial type lens ZCX . Note that under assumption (ii) all the eigenvalues of FJp) are zero except, possibly, the two f{ pl, p 2 } (p). Under assumption (ii) (as also under certain other conditions when n = l), Theorem 14 has been proved in Ivrii [1976a], while under assumption (i) in Iwasaki [1983a, 19841 and in Melrose [1983]. Under assumption (iii), it can be proved by combining the results of Ivrii [1976a] and Iwasaki [1983a, 19841. The proofs carried out in Ivrii [1976a] and Iwasaki [1983a, 19841 are based on the transformation of the operator P to an operator with "nice" lower terms by means of integropseudodifferential operators and on the energy estimates for the resulting operator. The proof of Melrose [1983] is based on the study of operators that are strictly hyperbolic for x, # 0 but degenerate at xo = 0; the Cauchy problem for the general operators is reduced to the one for degenerate operators by means of arguments based on strictly hyperbolic approximation and the uniqueness theorem. As a by product of the proofs, we have
165
Theorem 15. Suppose that the hypotheses of Theorem 14 are fulfilled. Let the inequality Ilm PW< ( L  n  2)lAp)L where f p w e nonzero real eigenvalues of F,(p), holds at any critical point then any pair (q m,q L ) will be a wellposedness index in K .
+
p E K x (IR"+'\O),
+
The following assertion well supplements Theorems 13 and 14.
Theorem 16 (Ivrii and Petkov [1974]). Let (X, 4) E X x (lR"+'\O) be a critical point of p. Zf FJX, 4) has nonzero real eigenvalues + p , then for the pair (1, q) to be a wellposedness index, it is necessary that IIm P"X,
< Cn(q + m  Olpl,
?)I
where C is an absolute constant.
Example 2. The operator P = D$+x~D;+
f:oj(D,?+xj'D;)+
j=l
n1
Dj' ( O < k < n  l , o j > O ) j=k+l
is regularly hyperbolic in IR"+l.
Example 3. The operator P+ = (0,"f xOD:)D, is regularly hyperbolic in R* x IR.
Example 4. The operators PI
=
Pz
=
0:
+ c wj(D,? + xj'D,") + 1
0,"
+ DOD1 + x ~ D ;+ 1wj(Dj' + xj'D;) + 1
k
n1
j=l
j=k+l
Dj' (0 < k
k
r1
j=2
j=k+l
(1 < k P3 = 0,"
< n  1, oj > 0), D,?
< n  1, oj > 0),
k
n1
j=2
j=k+l
+ xlDoDn + Df + 1 oj(D,? + xj'D,') + 1
Dj"
(1 < k < n  l , o j > O ) are x,hyperbolic in IRn+' but are not regularly hyperbolic in X if X n (x, =  . * = xk = 0 } # 0,where s = 1 for Pl, P3 ands = 2for P2. We now turn to Problem B.
V.Ya. Ivrii
166
11. Linear Hyperbolic Equations
1.5. Operators with Characteristicsof Constant Multiplicity Definition 10. An operator P is said to be an operator with characteristics of constant multiplicity (locally) in X if in the neighbourhood of every point p E { p = 0 } c X x (lR"+'\O) its principal symbol can be expressed in the form p = eq'. Here e ( p ) # 0, q is a real symbol, q ( p ) = 0, and Hq(p) # 0 is noncollinear with
a
5, and r E Z'\O and depends, in general, on the connected a5
component of the characteristic set { p = O}.
Definition 11. An operator P , x,hyperbolic in X,is said to be an operator with characteristic roots of constant multiplicity if all its characteristic roots 5, = Aj(x, 5 ' ) are of constant multiplicity rj E Z+\O, rl + *.. + r, = m (and then they depend smoothly on (x, 5 ' ) E X x (lR"\O)). Proposition2. (i) An x,hyperbolic operator is an operator with characteristic roots of constant multiplicity in X i f and only i f its principal symbol can be represented in the form (1.15) p = ao(x)py ... p p , where ao(x)does not vanish, rj E Z+\O, and where pi are strictly hyperbolic polynomials in X no two of which vanish simultaneously. (ii) The operator P , x,hyperbolic in X , is an operator with characteristics of constant multiplicity in X i f and only i f its principal symbol can be represented in the form (1.15), where p j are completely regularly hyperbolic polynomials in X no two of which vanish simultaneously.
=
'
B
Remarks. a) If condition (L) is satisfied in 8 x (lR"+'\O), it is also satisfied in
b) If qc0(p)# 0, we may assume, without loss of generality, that q = 5, A(x, 5') and Q = Do  A(x, D'),where A(x, D') is a pseudodifferential operator
q(x, dcp(x))= 0
(1.17)
in a neighbourhood of X and are such that dcp(X) = 4, and for any f E C" P(e'"0f) = O(pm') as p + +a
(1.18)
in a certain neighbourhood of X. The real functions cp satisfying (1.17) are known as phase functions.
Theorem 17. Let P be an x,hyperbolic operator in X having characteristics of constant multiplicity. For the Cauchy problem for P to be C"well posed, it is necessary (and sufficient i f X is a spatial type lens) that condition (L) holds.
(1.16)
x x (lR"+'\O).
with respect to x' with real principal symbol (which is a characteristic root of P). Then we may assume, without loss of generality, that in condition (L) A , are differential operators with respect to x, and pseudodifferential operators with respect to x' of order m  r. c) If P is xohyperbolic in X and qt0(p)= 0, then X E S , u S, qeoto(p)# 0 and Hq(p) # 0 is collinear with Hxo.In view of the Weierstrass basic theroem, we may then assume, without loss of generality, that q is a polynomial in 5, of degree 2, where the coefficient of 5: is 1, q is positive homogeneous in 5 of degree 2, Q and A,'s are, respectively, differential operators of order 2 with respect to x, and pseudodifferential operators of order m  r k with respect to x'. d) In view of Proposition 2, the x,hyperbolic operators that have characteristics of constant multiplicity and satisfy condition (L) can be characterized in the following inductive manner. Suppose that such operators have already been characterized in (1.15) for maxjrj r, and let p' = n j i Z , = , p j ,p = p'p''. Then P satisfies condition (L) if and only if it can be expressed in the form P = P'P" + A , where P' and P are differential operators with principal symbols p' and p", respectively, P" satisfies condition (L) and A is a differential operator of order m  r. e) Let dtq(p) # 0. Then condition (L) is satisfied in a neighbourhood of the point p if and only if for all (or for some, it is immaterial) real functions cp E C", that satisfy the eikonal equation
+
Thus for hyperbolic operators, Definitions 10 and 11 coincide if S , = S = 0 but Definition 10 is somewhat more general when S+ u S # Qr. Let us formulate the LeviStrangFlaschka condition that turns out to be a necessary and sufficient condition for wellposedness. Without loss of generality, we may assume that in Definitions 10 and 11 q is a homogeneous positive symbol in 5 of degree 1. Definition 12. An operator P with characteristics of constant multiplicity r in a neighbourhood of the point p = (X, 4) E X x (lR"+'\O) is said to satisfy the LeviStrungFlaschka condition (L) in the neighbourhood of this point if
where Q is a pseudodifferential operator of order 1 with principal symbol q and A , are pseudodifferential operators of order m  r, to within a pseudodifferential operator with symbol equal to zero in the neighbourhood of p (Taylor [1981], Tr6ves [1980]).
167
;
The necessity of condition (L) has been established by Flaschka and Strang [1971] and its sufficiency by Chazarain [1974] by constructing a parametrix for the Fourier integral operator. It was done under the assumption that P has characteristic roots of constant multiplicity. However, the necessity of condiion (L) in the general case of the operator having characteristics of constant multiplicity follows from this result in view of Remark a), and the sufficiency can be easily established, in view of Remark d), by means of the estimates (1.4), (1.11) and (1.12). For operators with characteristic roots of constant multiplicity, we need only the estimate (1.4). We note that when r 2 4 the proof of Flaschka and Strang [1971] does not go through unless condition (U,)is assumed to hold (see the note to the Russian translation of this work). But we can do away with this condition if we follow the method of Ivrii [1976b].
V.Ya. Ivrii
11. Linear Hyperbolic Equations
1.6. Irregularly Hyperbolic Operators of General Form. Let us formulate, first of all, the most general necessary conditions on the lower terms. The following assertion well supplements Theorem 13. Theorem 18 (Ivrii and Petkov [1974], Hormander [1977]). Let P be an x,hyperbolic operator in X with a real principal symbol p. Let p E X x (W""\O) be a critical point of p. Suppose that all the eigenvalues of F,(p) are pure imaginary. In order that the Cauchy problem for P be C"well posed, it is necessary that the following two conditions are satisfied: a) Im p S ( d = 0, b) IPS(P)lG 4Tr+Fp(p), where Tr+F,(p) = lpj1 and & ipj are all the nonzero eigenvalues of F,(p) (with regard to their multiplicity). The IvriiPetkovHormander conditions a) and b) turn out to be very precise. Theorem 19 (Ivrii [1977], Hormander [1977]). Let P be an x,hyperbolic operator in X , with real principal symbol p. Suppose that the following conditions are fulfilled: C) Z = { p = d p = 0} is a C"manifold; d) p E Z =. TpZ = Ker F,(p), Ker F i ( p )n Ran F,'(p) = 0, Spec F,(p) c ilR (then there exist E = f 1 and a zo(p) E Ker F J p ) such that o(zo, z) = 0, z 4 Ker F,(p) =.~o(F,(p)z, z) > 0). e) It is possible to choose a zo E C"(Z). I f the condition a) and the condition b') Ips(p)l < 4Tr F,(p) (in contrast to b), the inequality here is strict) are fulfilled, then the Cauchy problem for P is C"well posed in any spatial type lens z CX . Theorem 20 (Hormander [1977]). Let P be xohyperbolic in X . Assume that the following conditions, apart from condition c), are also satisfied: f ) p E Z * TpZ = Ker F,(p) c Ran(p) (that is, Z is an involutive manifold). €9 P E L  = > P Y P ) = 0. Then the Cauchy problem for P is C"well posed in any spatial type lens C X . Remarks. Theorem 19 remains valid under more general conditions (Ivrii [1977], Iwasaki [1984]). Under conditions of Theorems 19 and 20 very strong a priori estimates hold (Ivrii [1977], Hormander [1977]) that are established by the method of the divisor operator; in comparison to strictly hyperbolic operators, the loss in smoothness equals and 1, respectively. For model operators, we can prove that the Cauchy problem is C"well posed also under the conditions a) and b) (but not b')), but the loss in smoothness as compared to strictly hyperbolic operators will be 1. Under the hypotheses of Theorem 20, the condition g) turns out, in view of Theorem 18, to be necessary and sufficient for the Cauchy problem to be C"well posed. Example 4'. Let PI be one of the operators of Example 4, and let X ; n { x s = ..* = xk = O} # In order that the Cauchy problem for the Operator P, cD,,(c E (c) be C"well posed, it is necessary (and sufficient if X is a spatial type lens) that c E (c and IcI < G =
1.7. Necessary Conditions for the Cauchy Problem to be Well Posed. We do not have results, apart from Theorem 18, that are both general and precise enough. All the same, the following result is fairly general and in a number of cases proves to be precise enough.
168
Theorem 21. Let X E X ; , f = (0, ..., 0, 1) and q 8$,8p(X,
+
a.
cjoj.
. . .,q,,)E [O, 1)". Let
4) = 0
for
(1.19)
and
qoP(% B # 0.
(1.20)
If the Cauchy problem for P is C"well posed in X ; , then
a;ap,(X,
<) = 0
(1.21)
for lal
+ ( B  a, 4 ) < r(1  40)  (m  4.
By results a) and b) of Theorem 10, we have
+
4
= (q,,
169
Corollary. I f 4, is a characteristic root of multiplicity r, then in order that the Cauchy problem be C"well posed, it is necessary to impose conditions on all the members of P which are of order greater than m  yr, where y = i f E E 8 and y = 5 i f X E s+v s.
4
: b
In a particular case, Theorem 21 was proved by Ivrii and Petkov [1974]; for r > 4, it was assumed that condition (V,) holds. However, the method of Ivrii [1976c] enables us to prove it in its present form. By the same method, Mandai i recently established a more general theorem. In his formulation, Mandai We . note that both ! used Newton's polyhedra instead of halfspaces in Z+('"+') Theorem 21 and Mandai's theorem are noninvariant under the change of [ i coordinates.
I 1
1 \ \ L
Example 5. In order that the Cauchy problem for the operators
+ a,(x)D, + a,(x)D, + az(x) (0 E X : c a'), Pz = D; T xik+'x;'D: + bo(x)Do+ b,(x)D, + bz(x)
P, = D i
 xtkx:'D:
(OEX; c R * x lR) ( k , l E Z + )
' I
be C"well posed, it is necessary (and sufficient if X ; is a spatial type lens) that a , = x:'x;ii, and b, = xtx:6, with a",, 6, E C". 1.8. Degenerate Hyperbolic Equations. Among nonstrictly hyperbolic equations we identify degenerate equations that are either strictly hyperbolic or have characteristic roots of constant multiplicity in the exterior of initial or final or
V.Ya. Ivrii
11. Linear Hyperbolic Equations
intermediate hyperplanes. For a number of years only these equations were investigated, and there is a vast literature devoted to them. We mention one of the most specific results.
is the Weierstrass polynomial in xo and E(0) > 0. Let us expand the function t p ( x , )in Puiseux series for + x l > 0. Then
170
Re t,(xl)
=
C:~(+X~~'~(~), j>O
Theorem 22 (Olejnik [1970]). Let P = 0; 
171
5 DiaijDj+
i,j  1
bjDj j=O
where c:j E IR and p ( p ) E Z+\O. For an arbitrary function f (x),which is analytic in a neighbourhood of 0, we introduce the functions f,(x) = f ( x o Re t,(x,), x l ) and for them define the Newton polyhedra at the points (0, +O). To do this, we expand f, in series
+ c,
+
where aij = aji.Let the quadratic form n
a(x, 4') =
C i,
j=1
aijtitj
be positive definite for xo # To E [T, T+]. If the lower symbol b(x, 5') = satisfies the inequality Ib(x, <')I2
a
< co(xo  TO)'a(x, 8x0
and denote by N + (f,) the convex hull of the set
bitj
5 ' ) + W x , 5'),
Iff
then the Cauchy problem for P is Cwwell posed in any spatial type lens.
+ b(x)D1 + c(x),
P = D;  exp( 2x;')Df
= IR+ x
IR,
+ Bx;'+'+~( In x,)'
We now examine firstorder systems. In 0 1.10 and 9 1.11, P denotes a d x d matrix operator, A ( x , D') =
= ID,
 A(x, D')
2 aj(x)Dj+ a'(x), j=l
'
exp(  x;')D,,
0.
N,(X~E,,)c i~&i,) for all p = I, . . .,2m.
where the function 1 has a zero of infinite order at x , = 0. By Theorem 21, in order that the Cauchy problem for P be C"well posed, it is necessary that b has a zero of infinite order at xo = 0. But neither this necessary condition nor the sufficient condition following from Theorem 22 are precise.
Example 6 (Yagdzhyan [1980]). Let X
=
Theorem 23. In order that the Cauchy problem for P be C"well posed in some neighbourhood of 0, it is necessary and sufficient that B(x) = x:E(x), where is analytic and
More recent works on this topic are devoted to equations of order greater than 2 and to equations with infinite order of gluing of characteristic roots. A typical example of the latter is the operator P = 0;  A2(xo)D:
= 0, we set N,(f,)
and p(x, 5 ) = I t o  a(x, 5') and a(x, 5 ' ) are principal symbols of P and A, respectively, and g = det p.
1.10. Systems with Characteristic Roots of Constant Multiplicity. Let 5, = , A(x, 5 ' ) be a characteristic root with the following properties: ' a) In a neighbourhood of the point (X, 4') E X x (lR"\O) the root A(x, 5') is of constant multiplicity r. b) The matrix a"(x,5 ' ) = a(x, 5 ' )  IA(x, 5 ' ) is of a constant rank d  r + x in the neighbourhood of (X, 4'). Since for x = 0, the corresponding part of the Jordan's normal form of the matrix a is diagonal and it does not spoil the wellposedness, we should consider only the case x 2 1. Let q ( x )be a phase function, that is, a real function satisfying the eikonal equation j
and 0 < s, I, k E IR,B E C. A necessary and sufficient condition for the Cauchy problem to be Cwwellposed is { I > 0 or 1 = 0, k < 0 } if B 4 IR and { I > s or 1 = s,k
1.9. SecondOrder Equations in Two Variables. These equations are the simplest. Assuming the coefficients to be analytic and S+ u S = 0,Nishitani [1984a] carried out a complete analysis of Cwwellposednessof the Cauchy problem for these equations. Namely, let the operator P = D;  A(x)Df
+ B(x)Dl + c(x)Do+ R ( x )
be xohyperbolic in a neighbourhood of 0 (that is, A ( x ) 2 0), and let A(x),B(x) be analytic functions, with A(0) = 0 and A f 0. Then, in view of the Weierstrass ( x the ) neighbourhood of 0, where basic theorem, A ( x ) = x ~ ' ~ ( x ) E in
!
I
cp,,(x) = 4x5 dx,q(x)),
d X 4 3 =4
(1.22)
in a neighbourhood of X. Let us first examine the necessary conditions for C"wellposedness that were established by Petkov for r = 2, 3. Let R,(x, 5 ' ) and L,(x, 5') (k = 1, . . .,r  x ) be smooth vector functions that constitute a basis in Ker ii and Ker a"*, respec
V.Ya. Ivrii
11. Linear Hyperbolic Equations
tively; when x = r  1, we shall omit the index k = 1. Since the identity ( L , R ) = 0 is valid for x = r  1, r 2 2, we can find a smooth vector function h(x, 5 ' ) such that iih = R. If, moreover,
The case where only r = 2 is constant while x = 0, 1 may be variables has been discussed in Demay [1977]. Let ""p denote the matrix of cofactors of p, 1 = ps. ""p  1 { p, ""p},1' = { ""p.p s  1 { ""p,p } , where p s is the subprincipal
172
( L , ( x , d,,cp), P(R,(x, d , d ) = 0,
(1.23)
173
symbol of P, and let
a smooth vector function H ( x ) can be found such that d,,cp)H(x) If r = 3 and x
=
= P(R(x,d
x4).
denote the matrix Poisson brackets.
1, we may assume, without less of generality, that ( L j , R k ) =
8j28k2*
Theorem 24 (Petkov [1975]). Zf conditions a), b) and (1.22) are satiaied, r = 2,3 and x 2 1, then in order that the Cauchy problem for P be C"well posed, it is necessary that (1.23) is satisfied in a neighbourhood of X. Furthermore, for r = 3, x = 2, it is necessary that (L(x,d , , ~ ) ,P(h(x,dx,cp)) + H ( x ) ) = 0,
(1.24)
( L ( x , d x 4 , P ( H ( x ) ) )= 0
(1.25)
in a neighbourhood of Jz, while for r = 3, x = 1 it is necessary that 91.2.92,l
=0
(1.26)
Theorem 26. Let all the characteristic roots Aj(x, 5') of P be of constant multiplicity rj = 1, 2 in X x (IR"\O). Assume that there are smooth matrices sj(x, 5 ' ) and $(x, 5 ' ) such that 50
= Aj(X,
5'h
rj
=2
* l(x, 5 ) = P ( X ,
S)Sj(x, 5'1,
u x , 5 ) = g x , 5')P(X, 5).
(1.27)
Then the Cauchy problem for P is C"well posed in any spatial type lens. 1.11. Necessary Conditions for Regular Hyperbolicity of FirstOrder Systems in Two Variables. We conclude the present section with the only necessary condition, known to us, for the Cauchy problem to be C"well posed for systems with variable coefficients and with characteristics of variable multiplicity. Theorem 27 (Petkov and Kutev [1976]). Let n = 1. Let the operator
in a neighbourhood of E, where q
, k
=
( L j ( x ,d,cp), P(Rk(x,d,.cp))).
We can put these necessary conditions for wellposedness in an almost equivalent form as we did with the LeviStrangFlaschka conditions for scalar operators. Let x = r  1 or x = 1; in the latter case suppose that R , E Ran ii. We introduce Condition (L). For any real function cp satisfying (1.22) and any f E Cg there are scalar functions & ( X ) and vector functions vk(x)such that gk@) = 0 and
P = ZD, + a(x)D, + b(x) be x,hyperbolic in a neighbourhood of X E IR'. Let 3 be an eigenvalue of a@) having multiplicity r 2 2 with rank(a(Jz)  %) = d  1. Set p = (X, 1,1). For regular hyperbolicity of P in the domain X 3 Jz, it is necessary that the matrix Fg(p) has a pair of nonzero real eigenvalues. Otherwise, all its eigenvalues are zero, and for the Cauchy problem for P to be C"well posed in X ; 3 X, it is necessary that ( L , , a'R,)  ( L o ,a'R,)  2i(L0, b(Jz)Ro) = 0, where R,, R,, Lo,and L , are vectors such that
= O(p")
iiR, = 0,
as p + +a
in a neighbourhood of X. It can be demonstrated that condition (L) is equivalent to (1.23) for r = 2, x = 1, to (1.23), (1.22), (1.25) for r = 3, x = 2, and tn (1.23) and (1.26) for r = 3, x = 1; 91,2 = 0 or Y2,,= 0 in the neighbourhood of x. Theorem 25 (Petkov [1978]). Let all the characteristic roots Aj(x, <') of P be of constant multiplicity rj in X x (lR"\O). Let condition b) hold for any j , with xi E (0, 1, rj  11, and at each point (x, 5') E X x (IR"\O) let condition (L) hold for any j , with xj 2 1. Then the Cauchy problem for P is C"well posed in any spatial type lens.
0 2.
E*L, = 0,
iiRl
= Ro,
a*L1 = Lo,
Wellposedness of the Cauchy Problem in Gevrey Classes
2.1. The Main Definitions. As we have seen in the preceding section, the Cauchy problem is C"well posed for nonstrictly hyperbolic equations and systems only when, in general, some conditions on the lower terms are fulfilled.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
If we desire the Cauchy problem to be well posed for any hyperbolic operator with analytic coefficients, we must give up C"wellposedness and replace C" by some narrower class of functions but strive for this new class to be as wide as possible.
are noncharacteristic. Zf x > 1 and the Cauchy problem (1.1)(1.2) is G{")well posed, then P is xohyperbolic in X;.' Z however, the Cauchy problem (1.1)(1.2) is locally G{"}wellposed, then P is xohyperbolic in X;, where X is some neighbourhood of X,.
Definition 13. Let x E [l, co) and 1 E Z'U,. By an inductive Gevrey class G'(")(X)we mean the set of functions u E C!&(X) such that for any compact set KC X there are quantities R = R,, C = C, for which the following inequalities hold: ID"u(x)l < CRl"l(laI!)", Vx E K , a E Z+("+'): a, < 1. (1.28)
What is more, if { x } = (1) and the Cauchy problem (1.1)(1.2) is G(')well posed, then P is x,hyperbolic in X;. At the same time, by the CauchyKovalevskaya theorem, for { x } = (1) any noncharacteristic Cauchy problem is locally G(')well posed. On the other hand, the hyperbolicity is a sufficient condition for the Cauchy problem to be well posed in a suitable Gevrey class.
174
By a projective Gevrey class G'(")(X)we mean the set of functions u E C;:JX) such that for any compact set KC X and any R > 0 there is a C = C,,R for which the inequalities (1.28) hold. Evidently, GI(") c G'(")c G'("1) for x < x,, and G'i"} c G'l{"}for 1 > I,, and all the embeddings are proper (throughout, {. } will mean either or We denote by Grn(')and respectively, the space of analytic functions and that of entire analytic functions. (a)
Theorem 29 (Bronshtejn [1980]). Let P be x,hyperbolic in X , S+ = S = 0, and let X ; be a spatial type lens. Let x* = r/(r  l), where r is the maximum multiplicity of characteristic roots of P. If the coefficients of P are in G"(")(X)(G"(")(X))and x E [l, x*)(x E (1, x*]), then the Cauchy problem (1.1)(1.2) is G(")wellposed (G(")wellposed). Zf, however, x = x*, then this problem is locally G(")wellposed.
.))a(
Definition 14. Let the coefficients of the operator P belong to G'{"}(X).The Cauchy problem (1.1)(1.2) is said to be G'{"}wellposed if the following conditions are satisfied Existence condition a). For any f E G'{")(X;),g j E G(")(X,) ( j = 0,. . .,m  1) satisfying (1.1)(1.2). there exists a u E G'+'"{"}(X,+) Uniqueness condition b) of Definition 1. The Cauchy problem (1.1)(1.2) is said to be locally G'{"}well posed if the following conditions are satisfied Existence condition a'). For any f E G'{"}(X;), gj E G{")(X,), ( j = 0, . . ., m  1) there exists a neighbourhood X of X , and a u E G'+"{"}(X;+) satisfying (l.l)(l.2) in &+. Uniqueness condition b) of Definition 1 in each of the domains X' mentioned in the preceding condition. We shall examine the G(")wellposedness,G(")wellposednessand local G(")wellposedness. Since GI("),in contrast to GI("),are countablynormed spaces and since in conditions a) and a') above the requirement u E G'+"{"}(X;) is, apparently, meaningfully equivalent to the requirement that u E C"'(X:), the local G(")wellposedness has no new effects in comparison to the G(")wellposedness. Because till now we have not come across even a single example where the (local) G'{")wellposednessdepends substantially on I, we take some liberty and omit the letter 1 in the wellposedness notation. 2.2. Necessity and Sufficiency of Hyperbolicity. As we already mentioned, for the noncharacteristic Cauchy problem to be well posed in Gevrey classes the hyperbolicity is a necessary condition. Theorem 28 (Ivrii [1976b], Komatsu [1977], Nishitani [1978]). Let the coefi ficients of the operator P belong to Grn{"}. Assume that the hyperplanes {x, = t }
175
As confirmed by examples (see 9 2.3 and 9 2.4), in none of these assertions can the conditions imposed on x be, in general, relaxed. When S+ u S # fa, we have a less precise result. '
Theorem 30 (Ivrii [1975]). Let the operator P be x,hyperbolic in X and X ; be a spatial type lens. Let the characteristic symbol g of the operator be analytic in X x (IR"+'\O), and, finally, let the coefficients of P belong to G"{"}(X). Then the assertions of Theorem 29 remain valid for x* = (2r  2)/(2r  3). Remark. For r = 2, both the formulae for x* coincide. We now examine the following problem. Obtain conditions for the Cauchy problem to be well posed in a given Gevrey class. We shall assume below, without mentioning it each time, that P is x,hyperbolic in X and that X,+ is a spatial type lens. 2.3. Operators with Constant Coefficients. For such operators the wellposedness conditions can be easily obtained by means of the Fourier transformation. Theorem 31 (Hormander [1963]). Let P ( D ) be a (matrix) operator with constant coefficients and g ( N ) # 0, N = (1,0, ... ,0).Then for the Cauchy problem for P to be G(")wellposed, it is necessary and sufficient that for any a > 0 there exists a C = C, such that
vC'
E
IR"G(C,, 5')
= 0 => IIm
Col
< al<'ll/" + C,
(1.29)
where G(C) = det P(C). For the problem to be G(")well posed or locally G(")well posed, it is necessary and sufficient that (1.29) holds for suitable a and C. This result and the SeidenbergTarski theorem together give the following
176
11. Linear Hyperbolic Equations
V.Ya. Ivrii
in the neighbourhood of X for any real function larly, condition (L'J signifies precisely that
Corollary. Let g E hyp(N, M ) but G $ Hyp(N, M). Then the Cauchy problem for P is G(")wellposed for x E [l, x*] (and for these values only) and is G<")well posed as well as locally G("'well posed for x E [l, x*] (and for these values only), where x* = x*(C) is a rational number, x* 2 r/(r  1). When G E Hyp(N, M ) , we can take x* = 00.
C" and any h E C". Simi
(1.31)
+
in the neighbourhood of X for any real function E C" and any h E C". Finally, the implications ( L ) (L,) * (L'J (L,,) turn out to be valid for x > x', and it can easily be shown that of all these conditions substantially distinct are only the conditions (L),(Liij)with 1 d j < i d r, i, j E Z+.
Theorem 32 (Ivrii [1976b], Komatsu [1977]). Let the operator P be xohyperbolic in X and have characteristics of constant multiplicity. Let the coefficients of P belong to G"(")(X)(Gm(X)(X)), and let X: be a spatial type lens. Then for the Cauchy problem (1.1)(1.2)to be G(")wellposed (G(")well posed), it is necessary and sufficient that the condition (L,) (condition (LL))is satisfied in X;. For the problem to be locally G(")wellposed, it is necessary and sufficient that the condition (L',) is satisfied in X;', where X' is some neighbourhood of X,.
(1.29) Therefore for any N # 0, we can introduce the classes of polynomials HYPW, M , x) and =
E
P(exp i(qp + Ic/p"")* h) = O(p"'r(ll/x) a s p  +co
We note that the wellposedness conditions (1.29)may be replaced by the following conditions:
H ~ ~ (M N, ,(x))
+
177
n H ~ P ( NM, ,
X'CX
These classes decrease as x increases, and we can establish for them Properties (1.7)and (1.1 1).Theorem 7 can also be proved, but now the fundamental solution lies in gi,},the space of generalized functions over Gm{")n C,. But Theorem 31 will become valid only after the localization concept is modified. In what follows, we shall examine only scalar operators.
2.4. Operators with Characteristics of Constant Multiplicity. We can prove for such operators also necessary and sufficient conditions for the problem to be well posed in Gevrey classes. Suppose that in a neighbourhood of the point p = (X, 4) E X x (IR"" \O) the principal symbol of P is of the form p = aq', where
a
a@) # 0, q(p) = 0, and where Hq(p)# 0 and is noncollinear with t, r E Z'\O.
at
We shall say that the generalized Levi condition (L',) is satisfied in this neighbourhood if the representation (1.16)holds to within a pseudodifferential operator with symbol zero in the neighbourhood of p; here Q is a firstorder pseudodifferential operator with principal symbol q and A, are pseudodifferential operators of order not exceeding m  r k/x. If the orders of A, are strictly less than m  r + k/x for k 2 1, we shall say that the generalized Levi condition (L,) is satisfied. What is more, just as in 0 1.5, all the operators can be taken to be differential operators with respect to xo and pseudodifferential with respect to x'; Remark a) of 0 1.5 remains valid. If dcq(p)# 0 and rp is a real function that satisfies the eikonal equation in a neighbourhood of X and is such that drp(X) = f , then the condition (L,) means precisely that
+
P(exp i ( q p + +p'/")h) = p"'('l/,) exp i ( W X
(1
M!
'
+
+ +P"")
drp)(d+)"h o(1)) as p + +co
(1.30)
2.5. Necessary Conditions for Wellposedness in the Gevrey Classes. The following result is analogous to Theorem 21. i
1
, I
Theorem 33 (Ivrii [1976c]). Let the operator P have analytic coefficients. Let X E X ; , 4 = (0,. ., 0,1) and q = (qo, . . .,4") E [0,l)n+'. Assume that conditions (1.19)and (1.20)are satisfied. (i) If the Cauchy proboem (1.1)(1.2)is G'")wellposed, then
(1.32)
+
for la1 ( B  a, q ) < r(1  qo)  (m  s)x/(x 11, s c m. (ii) If qo = 0, X E X , and the Cauchy problem (1.1)(1.2)is locally G(")well ' posed, then i
d.pp,(Z,
4) = 0
(1.33)
+
for la1 ( p  a, q ) (r(1  qo)  (m  s)x/(x  I), s < m. (iii) If qo > 0, X E X , and the Cauchy problem (1.1)(1.2)is locally G(")well j piosed, then (1.32)holds. ' (iv) I f the Cauchy problem (1.1)(1.2)is G(")wellposed, then (1.33)holds. 1
i i
It follows from this theorem and from a), b) of Theorem 10 that for the Cauchy problem to be G(")wellposed or to be locally G(")well posed for X E X , (G(")well posed), it is necessary to impose suitable conditions on all the members of P of order not lower (greater) than m  yr(x  l)/x,where y = $ if X E 8 and y = 3 if X E S , u S.
Definition 15. Let the operator P have analytic coefficients. The Cauchy problem is said to be G(xlregular (locally G{"}regular)if it is G{")wellposed (locally G{"}wellposed) for the operaor P + Q, where Q is any operator, with coefficients in G"{"I, of order not greater than m  1.
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V.Ya. Ivrii
Example 8. Let P = Pl + cD, be the operator as in Examples 4 and 4,and 51. Let X, n {xl = = xk = O} # We recall that the condition that c E [ G, a] is necessary and sufficient for the Cauchy problem for P to be C"well posed. It can easily be shown that a necessary and sufficient condition for the G(")wellposednessis x < 2, while for the G(")wellposedness or for the local G(")wellposednessit is x < 2.
Theorem 34 (Ivrii  [1976c]). Let the operator P have analytic coefficients, and let X E X:, 5 = (0, ..., 0, 1) and q = (qo,. . ., 4.) E [0, l)"+'. Assume that d t a , 8 p ( ~t) , = Ofor la( + (fi  a, q ) < 0. (i) If the Cauchy problem (1.1)(1.2) is G(")iegular,then x < x* = a/(a  1). (ii) If qo = 0, X E X, and the Cauchy problem (1.1)(1.2) is locally G(")regular, then x < x*. (iii) If qo > 0, X E X, and the Cauchy problem (1.1)(1.2) is locally G(")regular, then x < x*. (iv) If the Cauchy problem (1.1)(1.2) is G(")iegular, then x < x*.
a.
c E C\[G,
Example 9 (Ivrii [19781). Let (i) 0 E X, and P
The following result has no analogue in the C"wellposedness theory.
z
P = 0:
t)= 0
where Z =
< 0,
(a, P ) #
(a, B),
= (o,O,. . . ,O), and that
a:oa;oP(%
= r/(r  1).
where p, rj E Z'\O, Lj # I, for j # k and 0 E X l . We set r = maxrj, x1 = r/(r  l), x2 = m/(m  1  1/p), where m = rl + ... + r,, and, finally, x* = min(x,, x 2 ) . If x1 < x 2 , a necessary and sufficient condition for the G(")regularityof the Cauchy problem is x < x*, while for the G(")regularity or local G(")regularity for 0 EX, it is x < x*. If, however, x1 > x 2 , then a necessary and sufficient condition for the G(")regularity or for the local G(")regularity for 0 E X, is x < x*; a necessary condition for the G(")regularity is x < x* and a sufficient condition is x < x*. If x = x*, the Cauchy problem is G(")well posed for any operator whose principal symbol is p and coefficients are analytic and depend only on xo. Finally, if x1 = x 2 , a necessary and sufficient condition for the G(")regularity or for the local G(")regularity for O E X, or for the G(")regularity is x < x*.
By Theorem 34 and the results a) and b) of Theorem 10, for the Cauchy problem to be G(")regularor locally G(")regularfor X E X&G(")regular), it is necessary that x < x*(x < x*), where x* = r/(r  2) if X E X and x* = r/(r  3) ifXES, US. Remark. If the corresponding inequality x < x* or x < x* does not hold in any one of the assertions (i)(iv) of Theorem 34 or of Theorem 35, then for the Cauchy problem to be suitably well posed it is necessary that Pml(X,= 0.
z)
2.6. Examples. Since fairly general results, apart from Theorems 29 and 30, concerning wellposedness in Gevrey classes of the Cauchy problems for operators with characteristics of variable multiplicity are not available, we confine ourselves to examining a few examples that are, in our view, quite instructive.
Example 7 (Ivrii [1978]). Let 0 E X, c IR2, and let P = 0:  x;"Df
+ a(xo)xGDl,
 (x,)~"D~
Example 10 (Ivrii [1978]). Let n = 1, and let
z) # 0.
If the Cauchy problem (1.1)(1.2) is G(")regular, then x < x*
+ a(xo)xGDl,
where v, 2p E Z ' , v < p  1, and a is analytic on [T, T,], with a(0) # 0. Then a necessary and sufficient condition for the G(")wellposedness,as well as for the local G(")wellposednessin case (i), is the inequality x < x* = (2p  v ) / ( p  v  l), while for the G(")wellposednessit is x < x*.
for la1 + ( B  a, 4 )
= D,Z  x;"Df
or let (ii) 0 E S , and
Theorem 35 (Irvii [1976c]). Let the operator P have analytic coefficients, and let X E X:, = (0, . .. , 0, 1) and 4 = (qo,... ,qn)E [0, 1)"". Assume that a;a,8p(R,
179
+ a(x)x;D1,
where p, v E Z ' , v < p, and a is analytic in X, with a(0) # 0. For the Cauchy problem to be locally G(")well posed or to be G(")well posed, a necessary and sufficient condition is x < x* = (2p  v ) / ( p  v). If p > 1, a necessary and sufficient condition for the G(")wellposednessis x < x*. If, however, p = 1, then a sufficient condition for the G(")wellposednessis x < x* = 2 and a necessary condition is x < 2. Thus the question whether the Cauchy problem for P is G(')well posed for p = 1 and v = 0 remains open.
1
Example 11 (Ivrii [1978]). Let (i) 0 E X, and
or let (ii) 0 E S , and
n (5;  q(X0)2"15'lZ)rd, S
p(x, 5 ) = 52
j=2
where 2p, rj E Z'\O, Aj # 1,for j # k and Ij> O V j . Then all the conclusions of the preceding example remain valid.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
Remark. If v 2 p in Example 7 or v 2 p  1 in Example 9, then the Cauchy problem is C"well posed. If in Examples 10 and 11 r = 1 and rn = 2, p = i, 1 or m = 3, p = *, then the Cauchy problem is C"regular. In all the Examples 7 to 11, the fact that the Cauchy problem is C"well posed (regular) implies that it is G{")wellposed (regular) provided that the coefficients of the operator belong to the corresponding Gevrey class.
where Char P = { g = det p = 0} c T*Q\O is the characteristic set of the operator P and m is the order of P. We start by examining the propagation of singularities of symmetric hyperbolic systems for which the Cauchy problem is undoubtedly well posed. Let
180
Example 12 (Yagdzhyan [1978]). Let 0 E X ,
P = D;  exp(  2x;')D:
= S
P = ID,  A(x, Dr),
181
(1.36)
where A is a differential or a classical pseudodifferential operator (Taylor [1976, 1979, 19811)with principal symbol a(x, tr); a(x, 5 ' ) is a Hermitian matrix.
and let
+ Bxb exp(  fix;')&,
Theorem 36 (Ivrii [1979c]). Let Q = (T, T+)x o and T E (T, T+). Let C(T, T+),9(o)), P u E Cm(Q)and #In, = u. Then
where B E C\O, 1 E lR,fi < 1. Then for fi c 1 and x c (2  fi)/(1  fi), and for fi = 1 and any x,the Cauchy problem is G(")and G(")well posed; at the same time, if fi = 1, it is C"well posed not for all 1 (see Example 6).
uE
WF(u)ln, c z'WF(v) u (N*Q,\O),
(1.37)
where I : Char PIn, + T*QT is the natural map, and N*Q, is the normal bundle of 0,. Moreover,
Example 13. Assume that fi = 0, 1 E Z+ in the preceding example. Then a necessary and sufficient condition for the G(")wellposedness,the local G(")wellposedness or the G(")wellposedness is that x < 2.
WF(u)l
c I'
WF(v),
(1.37')
if P is a differential operator.
tj 3.
;
Propagation of Singularities
; We mean by singularities various types of wave fronts (C", G{")or analytic). It should be kept in mind that the hyperbolicity of an operator is a concept that is local in x but global 5, while at the same time the wave front is a microlocal concept, that is, is local in both x and 5. Therefore for describing the propagation of singularities the hyperbolicity of the operator is not as important a property as are its other microlocal properties, such as microhyperbolicity. However, hyperbolicity is a strong constraint too (for instance, the localization of a hyperbolic operator is itself hyperbolic), and hyperbolic operators cannot belong to several microlocal types. We confine our attention to only those microlocal types that are admissible for hyperbolic operators.
3.1. Propagation of C"Singularities. We first recall the basic concepts of the microlocal analysis (Hormander [19711, Duistermaat and Hormander C19721, Taylor [1981], Trbves [1980]). The wave front of u is a closed conical set Definition 16. Let u E 9'(Q). WF(u) c T*Q\O: (X, 4) 4 WF(u) is there is a function 'p E Cr(Q), with p(X) = 1, and a conical neighbourhood r of such that
i
1'
(1.38)
t
B
11 ~
1i
for all s E IR, here F = Fx+<denotes the Fourier transform. For each s E IR,the same condition defines WFS(u),the wave front of order s. We recall that
'
WF(Pu) c WF(u) c WF(Pu) u Char P,
(1.35)
WF"(Pu) c WF"+'"(u)c WF"(Pu) u Char P,
(1.35')
Theorem 36 enables us to reduce the study of the propagation of singularities of solutions to the Cauchy problem to that of the propagation in the whole space. Suppose now that P = P(x, D)is a classical pseudodifferential operator of first order. Assume that the following is satisfied: Condition a). The matrix p ( x , t),the principal symbol of P, is Hermitian. Theorem 37 (Ivrii [1979c]). Suppose thut condition a) is satisfied. Let T*Q\O 3 V be an open conical set. Let 'p E C"(V) be a real positive definite homogeneous function (in 5 ) of degree 0 such that
(1.34)
(1 + ISI)SF('pu)E
The same assertions hold for WFs too.
then
WF(Pu) n V n {'p < 0 } n Char P = 0,
(1.39)
WF(u) n 8 V n ('p < 0 } n Char P = 0,
(1.40)
WF(u) n V n {'p < 0 } n Char P = 0.
(1.41)
The same conclusion holds for WF" too.
This theorem bears a strong resemblance to the uniqueness theorem for the Cauchy problem in a spatial type lens. But instead of the spatial type lens, a subset of lR:+', we now have V n {'p < 0 } , which is a subset of the phase space, and instead of the base of the lens we have the set a V n {'p < 0}, while the statements of the type u = 0 in X are now replaced by the statements of the type u 0 in W (that is, WF(u) n W = 0).

182
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V.Ya. Ivrii
In other words, the relations p' E K * ( p , V ) between the points p, p' E V are transitive and symmetric. We can now reformulate Theorem 37 in the following equivalent form.
Assume that the following is satisfied: Condition b). The condition (1.38) holds for cp = x,. Then for any real function the condition (1.38) can be rewritten in the equivalent form (HEH:okg)(PI .(H:og)l ( P )
V k = 1, ...,r(p) = dim Ker p ( p ) ,
'0
p E Char P n V,
Theorem 38 (Wakabayashi [1984]). Suppose that conditions a) and b) abooe hold. Let T*D\O 3 V be an open conical set and ji E Char P n K If
(1.42)
where
z
(H:og)(P) 09
E
(1.47)
WF(u) n K * ( p , V )n LJV= 0,
(1.48)
Like Theorem 37, this theorem also resembles the uniqueness theorem for the Cauchy problem, only now in a characteristic conoid the role of which is played by K * ( p , V). Theorems 36, 37 and 38 enable us to estimate from above the wave front of the fundamental solution of the Cauchy problem. Namely, let E = E ( x , y ) E C m ( X y9'(Dx)) , satisfy the conditions
P ( x , D,)E
TpT*Q: VC E f(p)a(z, C) 2 0 }
=I
~ ( x y),
SUPP E c {xo 2 Y o } .
be the dual cone, where a is a symplectic form.
TWF(E)c diag(T*X\O) u { ( p , p ' ) E (Char P n T*X\O)', p
E
K * ( p , V ) c K'(p, V ) ,
(1.44)
that is, for any Wneighbourhood of K'(p, V )we can find a W,neighbourhood of p such that p
E
W, n Char P => K'(p, V ) c W,
p " K*(p', ~ V )  ~ " EK * ( p , V ) , Vp, p'
E
V p'
E K'(p,
V ) o p E K'(p', V ) .
(1.45) (1.46)
(1.51)
K+(p', T*X\O)),
where T M = {(x, 5, y , q): (x, 5, y , q) E M } , M 2 = M x M and diag M 2 = { ( p , p), p E M } c M 2 is the diagonal of M 2 .
It turns out that Theorems 36,37, 38 and 39 remain valid for other types of operators, described in 91, too. Namely, suppose that one of the following , conditions holds. P is strictly hyperbolic or else has constant coefficients and is I Girding hyperbolic; the scalar operator P is completely regularly hyperbolic or else has characteristic roots or at least characteristics of constant multiplicity and satisfies the LeviStrangFlaschka condition or the hypotheses of one of the Theorems 4, 19 or 20; the firstorder matrix operator P satisfies the hypotheses of one of the Theorems 25 or 26. Then the conclusions of Theorems 36, 37, 38 or 39 remain valid if we carry out the following modifications. (i) In Theorem 36, we should now set %
P'P
(1.50)
Theorem 39. Let IR"'l3X be a spatial type lens and P a firstorder differential operator satisfyng (1.47) and (1.48). Then
We shall analyse below the notion of the generalized bicharacteristic in various particular cases. We can clearly take t = x, for a parameter on the generalized bicharacteristic. We now assume that T*Q\O 3 V is an open conical set and that PE Char P n K Let K * @ , V ) be the union of all generalized bicharacteristics issuing from p in the direction of increasing (decreasing)x, and lying in K It can easily be shown that (1.43)
(1.49)
If P is a symmetric x,hyperbolic differential operator of first order and R"+l3 X is a spatial type lens, then such a function exists, is unique and satisfies the dual system in y .
Definition 17. A generalized bicharacteristic of the operator P is a connected dP Lipschitz curve y = { p = p(t) E Char P } such that  E P ( p ) for almost all t . dt
K * ( p , V )are closed sets,
WF(Pu) n K * ( p , V ) = 0,
then WF(u) n K'(p, V ) = 0. The same conclusions hold for WF" too.
(a,"a!g)(P)= 0
< r = r ( p ) when conditions a) and b) are satisfied. for la1 + Theorem 37 can be stated in an equivalent form. Let p E Char P and let g p ( .), the localization of symbol g at p (see b) of Theorem lo), be a polynomial on TpT*Q. It can easily be shown that this polynomial is hyperbolic in the direction Hx0(p) if conditions a) and b) hold. Assume that T,T*Q =I f(p) = f ( g p ( * ) ,and Hx0(p)), containing H,,(p), is a connected component of the set {C, g,([) # O } . Under these assumptions (1.42) is equivalent to the inclusion H,,,(p) c f(p). Finally, let f"(p) = {Z
183
u = D&u(,,;
j = 0, . . ., WI  1.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
(ii) If P is hyperbolic only in X , then we should now regard u as an element of W ( 2 )and replace WF(u) by WF(u) n (T*X\O); this concerns Theorems 36 and 39. (iii) In Theorem 37, we replace condition (1.38) by (1.42). (iv) In the statements concerning wave fronts of finite order, we should write
Theorem 40. Let T*Q\O 3 V and V, be open conical sets. Suppose that g = egs in V, where e # 0, s E Z+\Oand g 1 is a real symbol of the principal type, that is,
184
Wp(pu),
5'Wp+mil
(uj) and WFs+mLl(u). Here
185
a
g 1 = 0 =E HBI# 0 and is noncollinear with l .
at
L = 0 for strictly and
(1.53)
Suppose further that P is a matrix operator such that g , ( p ) = Oadim Kerp(p) = s
j=O
completely regularly hyperbolic operators, L = f for operators satisfying the conditions of Theorem 19, L = 1 for operators satisfying the conditions of Theorem 20, L = r  1 for scalar operators with characteristics of constant multiplicity, L = x and L = 1 for operators satisfying, respectively, the conditions of Theorems 25 and 26, etc. It should be kept in mind that if we choose t = x, for the parameter, then the generalized bicharacteristic will be a Lipschitz curve only if x, E (T, T+).In order that the curve be Lipschitz near x, = T,, we must take for the parameter on it t = (T+  x,)"~ and t = (x,  T)'12,respectively. In conclusion, we remark that Theorems 37 and 38 remain valid also for more general operators than the ones mentioned above. In particular, the conditions imposed on the lower terms that guarantee the desirable propagation of singularities are far weak than the wellposedness conditions for the Cauchy problem (Ivrii [1979b1).
or P is a scalar operator satisfying the LeviStrungFlaschka condition or, finally, P is a firstorder matrix operator satisfying the hypotheses of Theorem 25 or 26. a) Zf V 3 y is a segment of the bicharacteristic of the symbol g 1 and WF(Pu) n y # 0, then either WF(u) n y = Qr or WF(u) 3 y. b) Zf V? y is a segment of the bicharacteristic of g 1 and the endpoints of y do not lie in V, c V, then we can find a function u E W(Q)such that WF(Pu) n V, = Qr and WF(u) n V, = ye n V,, where M, = {(x, At), (x, 5) E M , I > 0} is the conical hull of M . Therefore we have the following precise description of the wave front of the fundamental solution of the Cauchy problem. Theorem 41. Zf the hypotheses of Theorem 40 are satisfied in V = T*X\O and ifP is x,hyperbolic in X and X is a spatial type lens, then 'WF(E) = diag(T*X\0)2u { ( p , p ' ) E (Char P n T*X\0)2, p E y:(p')}. (1.54)
3.2. The Geometry of the Propagation of Singularities. In this section, we shall discuss the structure of the sets K*(p).We first note that if in a neighbourhood of p the characteristic symbol of the operator is of the form g = egy .. . g:, where e ( p ) # 0, gj are real and g j ( p ) = 0, sj E Z+\O, then cp satisfies (1.42) for g if and only if it satisfies the same for each gj. In particular, if C is a connected component of the set { g = 0} and if g = egs in a neighbourhood of C, e # 0 on C and p E C, then the sets K*(p), constructed for g and g,, coincide. If g ( p ) = 0 and gso(p) # 0, then (1.42) implies precisely that dcp > 0, where dxo ~
d ~
denotes the differentiation along the bicharacteristic of g, that is, the curve
dx, y = { p = p ( t ) }along which
(1.52) Therefore if the symbol g is strictly (or completely regularly) x,hyperbolic in X , then K * ( p )coincides with y$(p), the bicharacteristic of g issuing from p in the direction of increasing (decreasing) x,. Thus we obtain a unique description of the propagation of singularities for x,hyperbolic operators with characteristics of constant multiplicity. What is more, we can do away with the hyperbolicity condition and establish the following
A curve y containing its limit points and in a neighbourhood of which the hypotheses of Theorem 40 are satisfied and which is a bicharacteristic of the symbol g 1 will be called a regular bicharacteristic. A closed set M c V will be called a limiting bicharacteristic in V, C V if there is an open set V2, V, C V2 C V , and  a sequence of regular bicharacteristics Yk, with endpoints not in V,, such that =M n that is, if M n is the smallest closed set M such limk+m(Yk n that i&k+m (Yk n c M . From the arguments based on the Baire's category theorem it follows that assertion b) of Theorem 41 remains valid for limiting 1 bicharacteristic also. In many cases, this result enables us to show that Theorem i 39 is precise. It can easily be shown that if g is an x,hyperbolic symbol, then there exists a nowhere dense closed subset Z in C = { g = 0} such that the multiplicity of the characteristics of g is constant in C\Z, although the same depends, in general, on the connected component of C\Z. In view of Theorems 40 and 41, the main interest lies in the propagation of singularities in a neighbourhood of Z. We first assume that g can be factorized, that is,
v,)
v,)
vl
v,,
L
g
= egi'
.. . g p ,
(1.55)
where e # 0 and g j are real symbols of the principal type. Suppose that one of the following conditions holds: k=2,
91 = 9 2 = 0 * { g 1 , g 2 ) # 0 ;
(1.56)
V.Ya. Ivrii
186
k 2 3,
Vj g j = 0
11. Linear Hyperbolic Equations

complete analysis of the general situation (at a point) is still possible if the multiplicity of characteristics does not exceed 2. These are the situations that turn out to be the general case for narrower classes of systems of chrystaloptics and the theory of elasticity in anisotropic media. Let Z be a stationary set of g , that is, let
gj,to > 0
Suppose further that for any set X c { 1, . . .,k}, the facts that g j ( p ) = 0 V j E X and that 1= (lj)jEis a nonzero vector with nonnegative components imply CxA # 0, where (1.58) C x = ( { g i , gj})i,js;v (this condition will be fulfilled if, in particular, Vi,j
gi=gj=O,
i<j*{gi,gj)>O,
gj = 0 * gj,c0 > 0).
Then it can be shown that any generalized bicharacteristic of g is a polygonal bicharacteristic. That is, the curve y, parametrized by xo, consists of segments of bicharacteristics of g j , the endpoints (and only the endpoints) of these segments lying in C; at points of C a "transition" to bicharacteristics of other factors may take place; C n y is a locally finite set. Thus in the abovementioned cases, and in some other cases as well (Ivrii [1979c]), K' turn out to be "bicharacteristic trees" with branches at points of C. We note that in (1.57) and (1.58) we require definite signs and not merely nonvanishing of the quantities concerned. Otherwise there may take place infinitely many transitions on a finite interval for the polygonal bicharacteristics and there may appear generalized bicharacteristics lying entirely in Z and being bicharacteristics of the linear combinations of the factors (with nonnegative coefficients)but not of the factors themselves. If, however, the factors are in involution, that is, if gi=gj=O={gi,gj}=O Vi, <
< i,
Hgil,* . . HBit, 9
g '.I =
and
= gi, = 0 *
187
C = { d g = 0} c { g = O}.
(1.60)
To start with, we assume that at each point p E C the fundamental matrix Fg(p)has two nonzero real eigenvalues +@) (that is, g is effectively hyperbolic). Then it can be shown that for any point p E Z in some neighbourhood V of Z there exist two C"curves yj(p) = { p = pj(t)} ( j = 1,2) such that pj(0) = p, pj(t) # Z for t # 0 and such that y j is a bicharacteristic of 9 for t # 0 (but, of (O) course, this parametrization does not coincide with the one in (1.52)), and dpj dt is a nonzero eigenvector of Fg(p)corresponding to the eigenvalue ( l ) j p ( p ) . What is more, for a parameter on yj outside xo = T+ we may take t = xo. We denote by yf@) that part of yj(p) on which xo xo(p). Suppose, finally, that Cj = yj(Z)and Cj' = yF(C). Then we have the following results. Cj are closed conical sets, Zl n C, = Z, Zl u C, c C, and Cl u C2= C if and only if g can be factorized. If C is a C"manifold of dimension d, then Cj are C"manifolds of then y$(p) has no limit points in dimension d 1. If p E V n C\(CIT u CZT), C and K'(p) = y:(p); if p E C, then K'(p) = y:(p) u y:(p), and, finally, if p E ZT\C, then p E y?(p) for some point p E C and K'(p) = yj(p, p) u K*(p), where yj(p, p) is a segment of yj(p) between the points p and p. Thus under the above assumptions, generalized bicharacteristics are regular bicharacteristics not passing through points of Zand polygonal bicharacteristics having branches at points of C. [ We now assume that the hypotheses c)d) of Theorem 19 hold with p = g . We have the following results. ! a) If p E V n C\C, V is a neighbourhood of Z, then yg(p)has no limit points , in C, and, moreover, its distance from Z remains all the time of the same order;
+
1
(1.59)
@/at are linearly independent
and if g , ( p ) = = g k ( p ) = 0, then K'(p) are bicharacteristic sheets, that is, pieces of a kdimensional manifold Y 3 p spanned over the vector fields Hgj(j = 1, ...,k) and bound by (k  1)dimensional manifolds 3 p spanned over the vector fields Hgj(j = 1, . . .,k, j # i); for K'(p) we take the piece on which xo 2 x,(p). In the general case, the dimension of the bicharacteristic sheet K'(p) equals the number of factors that vanish at the point p ; the same factors also vanish on K'(p) but the other factors do not vanish on K'(p). Assume that the factors are divided into different groups in such a manner that factors belonging to the same group are in involution but those belonging to distinct groups are not. If { g i , g j } > 0 when g i belongs to the group with a smaller number than g j , then K'(p) are trees consisting of bicharacteristic sheets of different dimensions corresponding to distinct groups of factors. The geometry of the propagation of singularities becomes much more complicated when the characteristic symbol g cannot be factorized, and to analyse the behaviour of bicharacteristics becomes a far more difficult task. However, a
' 9
K*(P) = Y:(P). b) If p E C, then any generalized bicharacteristic passing through p lies in C
and satisfies the inclusion E
dt
P ( p ) n Ker Fg(p).
(1.61)
Let us replace condition e) of Theorem 19 by a stronger condition, namely, e') uIz has a constant defect d (then d 2 1). Then the manifold C is fibered into ddimensional bicharacteristic C"submanifolds A = A ( p ) 3 p : p E A TpA = Ran Fg(p)n Ker Fg(p);and in this case 01, = 0. Let p E A . Then the restriction of the localization g p ( . ) to Ker F:(p) is a well defined quadratic form on the factor space KerF:(p)/Ker Fg(p). Since there exists a unique isomorphism between this factor space and T*A,
V.Ya. Ivrii
11. Linear Hyperbolic Equations
we obtain a symbol h defined on T*A which, by statement b) of Theorem 10, is strictly x,hyperbolic of second order. Then K'(p) lie in A ( p ) and are conoids of dependence for h. In particular, in the general case d = 1, the bicharacteristic manijiolds (they are the limiting bicharacteristics as well as the generalized bicharacteristics) are the curves y = { p = p ( t ) } defined by the inclusions
E ( x , y ) is a Fourier integral distribution in the sense of Melrose and Uhlmann [1979b], that is, is a Fourier integral distribution with a Lagrange manifold having selfintersections. For scalar operators with characteristic roots of constant multiplicity that satisfy the LeviStrangFlaschka condition a parametrix was constructed in Chazarain [19741 by means of Fourier integral operators, while for firstorder systems satisfying the hypotheses of Theorem 25 in Petkov [1978]. For various classes of effectively hyperbolic operators the parametrix has been constructed in Kucherenko and Osipov [1983], Alinhac [1978], Yoshikawa [1977,1978a, b, 19801. For operators with factorizable principal symbols for the case when the factors are in involution and also for operators satisfying the conditions of Theorem 20, parametrices have been constructed in Sjostrand and Uhlmann [1979a, b]. For diagonalizable firstorder systems the parametrices have been constructed in the form of series in Kucherenko [1974]. A more detailed study led Kumanogo and his students to the construction of the theory of Fourier integral operators with multiphase functions (Kumanogo and Taniguchi [19791, R. Lascar [1981]).
188
dP
E
dt
Ker F,(p) n Ran F,(p).
(1.62)
Since now through every point p E Z there passes exactly one generalized bicharacteristic, assertions a) and b) of Theorem 40 hold for generalized bicharacteristics in this case and the inclusion (1.51) turns into an equality. Now there arises the question as to how precise is the description of the propagation of singularities obtained above. We have in mind the inclusion (1.51) in the first place. The existing results (Ivrii [1979a, b], Uhlmann [1982]) are not exhaustive nor even somewhat general, but enable us to presume that in the general case (1.51) turns into an equality for almost all lower terms, although for some lower terms there may be abnormally poor propagation of singularities. What is more, if g can be factorized into two factors, not in involution, then for almost all lower terms all the singularities, and not only the typical ones, are propagated in a prescribed manner (Ivrii [1979a]).
3.3. The Construction of a Parametrix. The length of the present survey paper does not permit us to give an account of the construction of a parametrix for various cases, and we confine ourselves to a cursory look at the basic results. For strictly hyperbolic operators and for diagonalizable hyperbolic systems of first order, with characteristic roots of constant multiplicity, the parametrix was constructed by many authors by means of the oscillatory integrals of a certain type. The investigation of these integrals and operators led Hormander [1971] to the development of the theory of Fourier integral operators (and distributions). Similar reasonings connected, however, not with the construction of the parametrix but rather with the global construction of individual asymptotic solution of the Cauchy problem with oscillatory initial data had earlier led Maslov [1965, 1973, 1976, 19833 to the construction of a canonical operator. The further development of these two theories ran parallel to a great extent, although each one of them has its own deep individual features. It should be noted that not the fundamental solution E ( x , y ) is a Fourier integral distribution in the strict sense but a distribution U(x, y ) such that P(x, Dx)U = 0, DiUlxo=yo = iSj,m16(x y ) l
( j = 0, ..., m  1).
(1.63) (1.64)
If the coefficient of Dg in P is 1, then (1.65)
I t
i
189
3.4. Propagation of Analytic Singularities and Gevrey Singularities. We first define the wave fronts WF{")(u),x E [l, a), corresponding to the Gevrey classes.
Definition 18. Let u E 9'(sZ). Then (X,4) 4 WF(")(u)if there exists a function r of 4 such that the inequality cp E G'")n Co(sZ),with cp(X) = 1, and a concial neighbourhood
IF(cpu)(t)(< C exp(AItl"")
for
5 Er
(1.66)
holds for some C and A > 0. Similarly, Definition 19. Let u E 9'(Q). Then (X,4) 4 WF(")(u) if there exists a function G ( " )n C0(Q),with q(X) = 1, and a conical neighbourhood r of 4 such that (1.66) holds for any A > 0 and suitable C = C,. cp
E
The extension of these definitions to the case x = 1 is not automatic. For distributions there are three equivalent definitions of the analytic wave front (or of the essential spectrum or of singular spectrum); these are by Sato, by Bros and Iagolnitzer, and by Hormander. Here we present the one given by Bros and Iagolni tzer. Definition 20. Let u E 9'(sZ). Then (X,4) 4 WF,(u) = WF(')(u) if there exists a function cp E C,"(sZ),cp = 1 in a neighbourhood of X,and a conical neighbourhood I' of 4 such that the inequalities IF(cpu)(x,<,211 < CN(l
+ I
v t E r,
1 E [O, Yltll, (1.67) hold for suitable constants A > 0, y > 0 and CN,and for all N E Z ' ; here (Fu)(x,t, 4=
s
exp(i < y , 5 )

Alx

y12)u(y)d y
191
V.Ya. Ivrii
11. Linear Hyperbolic Equations
is the FourierBrosIagolnitzer transformation. We note that WF{")(u) and WF(u) can also be defined in terms of this transformation.
tinct variants, namely, in a strong sense and in a weak sense. As we already remarked, the wellposedness question for the mixed problem is far more complicated than the one for the Cauchy problem, because it depends on three objects. Namely, on the operator P itself, on the boundary values and on the boundary surface itself. What is more, the class of admissible operators P gets somewhat enlarged because on the other side of the boundary surface or below the initial or above the final hyperplanes the operator may fail to be hyperbolic. Let o be an open domain in IR" with a C"boundary do. Let X = [T, T+]x 0 be a cylindrical domain with respect to x , with lateral boundary S = [T', T+] x do.Let v(x)denote the inward normal to S and S + = { T,} x 0 the base of X , m < T < T+ < +a. Put N = dx,. We shall represent ao in a local coordinate system by the equation x , = 0 and w by the inequality x , > 0. We examine the following mixed problem:
190
It can easily be shown that WF{")(u)are closed conical subsets of T*SZ\O, and that WF(")(u)3 WF("')(u) 3 WF("')(u) 3 WF(u) for 1 < x < xl. What is 'more, WF{")have basic properties of the usual wave fronts: n,WF{")(u) = sing SUPP{X)(4,etc. For x > 1, the propagation of singularities for hyperbolic operators was investigated by Wakabayashi [1983a]. The propagation of analytic singularities was studied by many authors, but, in our view, Wakabayashi [1983b] succeeded in putting these results for hyperbolic operators in the most convenient form.
Theorem 42. Let P be an x,hyperbolic operator in SZ whose coefficients lie in G"{')(SZ). Assume that at least one of the following three conditions is fulfilled: (i) x* = r/(r  l), where r is the maximum multiplicity of the characteristic roots of P . (ii) P is a scalar operator with characteristics of constant multiplicity and the condition holds. (iii) The coeflicients of P are constant and (1.29) holds with x = x*. W e have the following results. (i) If { x } = (x), x E [l, x*] or if { x } = ( x ) , x E (1, x*], Pu E G"{"I(SZ) and v = (DiuInT, j = 0, . . .,m  1) are the Cauchy data, then (1.37") WF{")(u)l,, c zl WF{")(u); (ii) If { x } = (x), x E [l, x*) or if { x } = ( x ) , x E (1, x*], p T*S2\0 3 V is an open conical set and WF{")(Pu)n K * ( p , V ) = fa,
E
Char P n V, and
WF{"}(u)n K * ( p , V )n d V = fa,
Pu = f inX,+, (2.1) B k U = (pk ( k = 1, . . . ,p) On s;, (2.2) ( j = 0, ..., rn I), D&Ixr = gj (2.3) where P and Bk are scalar differential operators, with C"coeficients, of order rn and mk, respectively, 0 < mk < m  1. We also consider the mixed problem Pu = f
in X,+,
Bu = cp on S:,
(2.1') (2.2')
(2.3') where P is a d x d matrix differential operator of first order with C"coefficients and B = B ( x )is a p x d C"matrix. Thus right from the beginning we exclude from our study those cases where the lateral surface has corners or edges or where different types of boundary conditions are prescribed on its different parts. Before defining a C"well posed mixed problem, we remark that for no mixed problem with p > 0 (that is, other than the Cauchy problem) does infinite smoothness of the righthand side, the boundary and initial data imply infinite smoothness of its solution unless compatibility conditions of infinite order are satisfied. Namely, we say that Ulx, = 9,
( 1.47') ( 1.48')
then WF{")(u)n K * ( p , V ) = fa.
Corollary. Let P be a scalar operator with characteristics of constant multiplicity in SZ and with coefficients in G"{"}(Q). Let P satisfy the condition (L;*) with either { x } = (x), x E [l, x * ) or { x } = ( x ) , x E (1, x * ] . Then the assertions a) and b) of Theorem 40 remain valid for WF{"}too.
Chapter 2 Mixed Problems for Hyperbolic Operators
{ f, cpk(k = 1, .. ., p), g j ( j = 0, .. ., m  l)}E C" satisfy compatibility conditions of infinite order if there is a u E C" such that (2.1)(2.3) are satisfied to within functions having zero of infinite order for x , = T.
5 1. Wellposedness of Mixed Problems
Definition 1. The mixed problem (2.1)(2.3) is said to be C"well posed if the following two conditions are satisfied: a) The problem (2.1)(2.3) has a solution u E Cm(X,')for any
1.1. Preliminary Remarks. In the present section we consider the question of the wellposedness of mixed problems. By wellposedness we mean the Cmwellposedness as well as the L,wellposedness in two of its substantially dis
{f,(Pk(k = 1, ..., p), g j ( j = 0, ..., m  l)} E C" that satisfy the compatibility conditions of infinite order.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
b) The fact that u E C" satisfies, for any t > T, (2.1)(2.3) with go = = gml = 0 and with f = 0, (pk = 0 ( k = 1, . ..,p) for xo < t, implies that u = 0 for xo < t.
that is, for the solvability of the problem the number of boundary conditions must not be too big. We remark that the adjoint problem does not always exist. However, if the original problem is, for instance, normal (see below), the adjoint problem does exist. Similar arguments apply also to the problem (2.1')(2.3')but now (2.7) and (2.9)are to be replaced, respectively, by
192
For the problem (2.1')(2.39, the definitions of compatibility conditions of infinite order and of C"wellposedness are analogous. In both cases, we shall assume that the hyperplanes {xo = t } are noncharacteristic, that is, g(x, N ) # 0
vx E x ,
(2.4)
where g is the characteristic symbol of P. It can easily be shown that for the mixed problem (2.1)(2.3)(or (2.1')(2.3')) to be C"well posed it is necessary that P is xohyperbolic in X;; the proof coincides word for word with the corresponding proof for the Cauchy problem. Therefore we shall assume that P is xohyperbolic in X. Then for x E S, 5 E IR"+', z E C \ R , the polynomial g(x, 5 zN av(x)) (in a) has no real roots a. What is more, the degree of this polynomial, by b) of Theorem 10, is independent of (5, z) E IR"+' x ((C \R); we denote it by M(x). Thus the number of roots o of this polynomial with Im o 2 0 is also independent of (5, z) E IR"+l x ((C\lR); we denote it by M'(x), and set M+(x)+ M  ( x ) = fi(x). Moreover, if M is independent of x E S , then M*(x) depends solely on the connected component of S. Using the method of Ivrii and Petkov [1974], we can easily prove that for the a priori estimate, which is a consequence of the C"wellposedness of (2.1)(2.3) or of (2.1')(2.30, to hold it is necessary that
+
p 2 M+(x)
at every point x E S,+.
(2.5)
P*u= h inX,+, = ll/k
( k = 1, ..., p')
(2.6) On S:,
suppucXn{T<x,
(pu,u)X
 (u, p*')X
=
c
(Bk%
bku)s

i
y
Ii
I
I
!
(2.8)
:
c
( Y k U , cku)s
(2.9)
k=l
for any u, u E Cm(X),with DAuls = 0 and Diul,+ = 0 ( j = 0, . . . , m  1). Here (., .)M denotes the scalar product in L,(M), and Bk and y k are certain differential operators of order m  mk  1 and m  m;  1, respectively, in which mi is the order of Ck.From this a priori estimate one can deduce by the same methods the inequality p' 2 M(x),
x E s,,
(2.5')
(2.10)
If in a neighbourhood of X E S; rank p ( x , v(x))is constant and equals s, then in this neighbourhood the adjoint problem is defined with p' = d  s  p, and (2.5),(2.5')imply
P'
k=l
 (YU, C 4 ,
rank B(x) = p, Vx E S;.
'
P
S
(2.7')
where C , B and y are, respectively, p' x d, p x d and p' x d matrices. Evidently, a necessary condition for (2.1')(2.3')to be C"well posed is that
(2.7)
that is, for the problem which is such that
on S,+,
(Pu, 4 x  (u, p*4, = (Bu,P 4
+
If the number of boundary conditions is too small, then an asymptotic solution violating this estimate is constructed. On the other hand, the fact that the problem (2.1)(2.3)is solvable implies a priori estimate for the adjoint problem
CkU
Cu = ll/
193
,
M + ( Z )< p
< M+(Y)+ d  s  A?@).
(2.5")
Hence, if fi(x) = d  s (which, for instance, holds when S is noncharacteristic), we obtain p
= M+,
(2.5"')
a condition that is necessary for the problem (2.1')(2.3') to be C"well posed. We know that the mixed boundary problem for an equation can be reduced to that for a firstorder system, and, moreover, since the noncharacteristic mixed boundary problem (that is a mixed problem with a noncharacteristic lateral boundary) is reduced to a similar problem having the same number of boundary conditions. In this reduction the C"wellposedness of the original problem is equivalent to the C"wellposedness of the resulting problem. Therefore (2.5'")is a necessary condition, under the assumption that the boundary surface is noncharacteristic, for both the problems (2.1)(2.3)and (2.1')(2.3')to be C"well posed. And all the arguments remain valid provided that the solution is required to exist and to be unique in the same class of functions; for instance, in C'(m < 1 < 00). Unfortunately, the arguments of this section have not been carried out accurately in any of the works known to the present author. In the case of characteristic boundary, the number of boundary conditions may depend on the lower terms of P and on the required smoothness of the solution, as, for instance, for the operator P = 002  X1D:
+ aD1,
u E (C.
(2.11)
V.Ya. Ivrii
11. Linear Hyperbolic Equations
In order to overcome this difficulty, the concept of a uniformly characteristic boundary is introduced. A surface S = {'p = 0}, with a 'p # 0, is said to be uniformly characteristic if for small a the surfaces {'p = a } are characteristic, and of the same multiplicity. This concept is very important on account of the Maxwell system.
We assume that p = M', that is, that (2.15) is a square system. By the Lopatinskij determinant of (2.1)(2.3) we mean the determinant of (2.1 5):
1.2. Operators with Constant Coefficients. In this and the next section, we follow Sakamoto [1970a, b, 19803. In the present section P = P ( D ) and Bk = Bk(D) are operators with constant coefficients, X = IR" x K+ = {x E IR"+', x, 2 0} is a halfspace and S = IR"= {x E IR"", x, = 0} a hyperplane. Let the operator P ( D ) be Girding x,hyperbolic. This is a necessary and sufficient condition for the C"wellposedness of the noncharacteristic Cauchy problem, and, hence, a necessary condition for the C"wellposedness of the mixed problem. We assume that the hyperplanes {x, = t } are noncharacteristic. Then the polynomial G (to,5", 5") has no real roots 5. for 5'' = (t1,..., t,,) E IR"' and for Im to <  C; recall that G(5) = det P(5). Assume that the hyperplane S is noncharacteristic, that is, g(v) # 0, where v is the inward normal to S . Then, for 5'' E IR"' and Im to( C, G(t,, t",5,) has M' roots t,,with Im 5, 3 0. Denoting these roots by A:(, 5") ( j = 1, ...,M') and setting
We can define a more general determinant 9 ( q TO), where q = (to, 5") E IR" and 0 belongs to the hyperbolicity cone T(g,N ) of P(D).
194
n ( tn
M'
G,(O
=
j =1
 A,?(<,,
we have G ( 5 ) = cG+(t)G(<), c # 0. We perform (formal, for the time being) Fourier transformation with respect to X" + 5'' E IR"' and FourierLaplace transformation with respect to xo + to, Im to< C. Then (2.1)(2.3) goes over to a boundaryvalue problem for an ordinary differential equation with constant coefficients: (2.12)
Bk(50, t",Dn)alxn=O = @k ( k = .  * ) (2.13) where the righthand side F of (2.12) is determined by the righthand side of (2.1) and by the initial data, while @ k are determined by the boundary conditions and the initial data. The general solution of (2.12), with F = 0, that decays as x, + +a is of the form =
M+ s C 2x1 cj
eiXnrn<','G;'(t)d t , ,
t")= det
(&
Bk(t)ti1c;'(5)
dtn) k ,j = l ,
..., p
.
+
Theorem 1. For the problem (2.1)(2.3) to be C"well posed, it is necessary that 9 ( q + z N ) # 0 Vq E IR", Im 7 < C,
(2.16)
with a suitable C, . In particular, if P ( D ) = p(D) and Bk(D) = bk(D)are homogeneous operators, it is necessary that
l(r]
+ 7N)# 0
vr] E IR",7 E C\IR,
(2.17)
where 1 is the Lopatinskij determinant for { p , b,, .. .,b,,}. For nonhomogeneous operators also we can define 1 by its principal parts. We assume that 3q E IR": l(q  iN) # 0.
(2.18)
Then 1 is the principal part of 9 (If I(  iN) = 0, then 1 is defined precisely as the principal part of 9.)
t")),
P(t0, t",Dn)a = F,
9(tO?
195
(2.14)
j='
where y is a closed contour in C that goes round all the roots A:(tO, ("1 ( k = 1, . . ., M + ) once in the counterclockwise direction and cj are arbitrary numbers. Substitution of (2.14) into (2.13) yields the following system of algebraic equations:
Theorem 2. For the problem (2.1)(2.3) to be C"well posed when (2.18) holds, it is necessary that 1("
# 0.
(2.19)
The proofs of Theorem 1 and 2 are based on the construction of asymptotic solutions. It can easily be shown that (2.16) and (2.19) imply (2.17). Theorem 3. If (2.16) and (2.19) hold and the boundary conditions are normal, that is, if j > k*mj > mk and
bk(v)# OVk,
(2.20)
then the boundary value problem (2.1)(2.3) is C"well posed.
Assume that (2.16) and (2.19) hold. Let T,be a connected component, containing N , of the set (0 E r(g, N ) , I(  i0) # O}. Then
r, is an open convex cone,
+ 70) # 0 9(r] + 70) # 0 l(r]
(2.21)
< 0,
(2.22)
= C,
(2.23)
V q E IR", T: lim z
31 E IR,
7: Im
z
(compare with the similar properties in 4 1.3,Chap. 1).
11. Linear Hyperbolic Equations
V.Ya. Ivrii
196
holds with the constant independent of u, to,t or y, where
Finally, let
r+ = r ( g , N ) n ( 5 + OV,r E r,,0 E W} and let r; be the dual cone and'r = r' n { x , 2 O}.
Ilu; Kl14,y=
PEj = 0 for SUPP
and K(t,,,t)= K n {to < xo < t}. The other norms Ill'; Srlllq,7and 111.; S(tO,t)JJ/q,yare defined in a similar fashion, only that the derivativesare now taken only along S. It is easy to see that if the boundary is noncharacteristic, then the estimate ' . Moreover, for s = 0 analogous estimate (2.25,) implies (2.25,) for all s E Z is obtained if one employs the norms used by Sakamoto [1970a, b, 19801. Namely, Ilu; Kllb,y= Ileyxou;Kllq, and similarly, for Illu; KIllh,Y;the index y = 1 is omitted. Let 9be the set of infinitely smooth righthand sides and initial and boundary data having compact supports and satisfying the compatibility conditions of the closure of % in the space infinite order. Let FS,
x , > 0,
Ej c {xo 2 O},
(2.24)
BkEjlX.=, = d i j s ( X 0 ) d ( X r r ) (k = 1, ..., p), then supp Ej c r:. The proofs of Theorems 3 and 4 are based on the application of Fourier transformation in the complex plane with respect to the variables (x,, .. .,x ,  ~ ) and on proving a Holmgren type uniqueness theorem. In the latter, the fact that the problem is normal plays an important role; this guarantees that the adjoint problem, satisfying all the conditions of Theorem 3, exists. Theorem 4 and Theorm 7 of Chap. 1 together describe the support of the solution to the problem (2.1)(2.3) All these arguments apply also to the problem (2.1r)(2.3r)with the only difference that the condition (2.20) is replaced by (2.10) and the inequalities 9 ( q z) # 0 and l(q + T O ) # 0 by the conditions that the row matrices
+
+ + +
B " O P ( ~ T e + O V ) (mod G+(q B"O~(? ze BV) (mod g+(q
+ + av)), + + OV))
are, respectively, linearly independent; here "'(. ) denotes the matrix of cofactors.
1.3. Strong L,wellposedness of the Mixed Problem (for Equations). As in the preceding section, here also we follow Sakamoto [1970a, b, 19803. The strong L,wellposedness (Hwellposednessin the terminology of Sakamoto) is the strongest form of wellposedness.We denote by Hsvarious Sobolev spaces. We shall assume in what follows that either ao is compact or ao = { x , = 0 } and that all the coefficients of P and Bk are constant for lxrl > R. For simplicity, we assume that T+ co.
=
Definition 2. The mixed problem (2.1)(2.3) is said to be strongly L,well posed if it is C"well posed and if, for all u E C", all y 2 0, to, t E [T, T+], to <: t and all s E Z ' , the a priori estimate X(t,,r)IIml+s,y+
IIU;
XrIIml+s,y +
IIU; S(to,t)IIml+s,y
1 y4t"111eY"~D"u; L2(K)ll, I4cq
Theorem 4. Zf the hypotheses of Theorem 3 are satisfied and E j e Cm(E+,9(lR")) is a solution of the problem
J~IIU;
197
be the set of righthand sides, initial and boundary data having smoothness as mentioned and satisfying the compatibility conditions of orders. It can be shown that if S is noncharacteristic and the boundaryvalue problem is normal, then the compatibility conditions are of local nature, and when ao = {x, = 0}, the compatibility conditions of order s are the p ( s + m  1)  E m k conditions
(
k
)
that connect the values on S , of the function DzD.""f(ao + a,, < s  l), D2gj(an< s + m  2  j ) , Dz@'(txo < s + m  2  mk) and their derivatives along S,. These conditions enable us to express @q&, in terms of the remaining functions and their derivatives along s,. It follows from the definition that if (2.1)(2.3) is strongly L,well posed and if t {f,g j ( j = 0, . . . m  I), pk(k = 1, . . . PI} E E , I Y then the problem (2.1)(2.3) has a solution I I I
n
s+m1
uE
j=O
Cj([T, T,],
Hs+"''j (0)) c HS+m'(XT+);
1 moreover, D"u E HS+mll"I(Sg) for all a, la1 < s + m  1. +
Theorem 5. Let the estimate (2.25,) hold for some s. Then (i) P is strongly x,hyperbolic in X ; . (ii) Either at no point is S; noncharacteristic or else p = m, s = 0. (iii) p 2 M'. (iv) Zf p = M + , then the uniform Lopatinskij condition holds: I(x, 5
+ TN) # 0
vx Es ,; 5 E IR"+l, I(x, 5 ) # 0 v x E ST+, where I(x, l )= I(x, 5  OiN).
z
(2.26) E
(c\IR,
5 E IR"+l\lR.V,
(2.27)
11. Linear Hyperbolic Equations
V.Ya. Ivrii
198
199
L,wellposedness. The first of these is of the form
Conversely, we have Theorem 6. Let the operator P be strictly x,hyperbolic in X , S be noncharacteristic and let p = M'. If the uniform Lopatinskij condition holds on S , then the estimate (2.25) is true. The proof of Theorem 6 uses the method of the divisor operator, which in the case of a boundaryvalue problem turns out to be much more complicated than for the Cauchy problem. On account of (2.27) and (2.22), N in condition (2.26) may be replaced by any vector 0 E T(g,N). What is more, if the boundaryvalue problem is normal and if the hypotheses of Theorem 6 are satisfied, then the same conditions are fulfilled for the adjoint problem. Therefore, by means of functionalanalytic arguments, we can establish the following
or Ilu; Xtlls + Ilu; S(t0,t)llSG CClIu; Xt0lls
IIPu; &lIs
dt'
+ IIIBu; S(t0,t)lllSl (2.28;)
Theorem 8. Let the estimate (2.25,) hold for some s. Then (i) The operator P is x,hyperbolic in X: and the matrix p(x, 0, 5') is uniformly diagonalizable in X ; (it is assumed throughout this section that p(x, N ) = I). (ii) Either at no point is S; noncharacteristic or else p = d and s = 0. (iii) p 2 M'. (iv) If p = M', then the uniform Lopatinskij condition holds: a) For all X E S:, 5 E IR"+l, and r cC\lR, the rows of the matrix BCop(x,5 r N av) are linearly independent (mod g+(x,5 rN + ov)). b) For all x E S:, k E lR"+l\lR*v,the rows of the matrix Bcop(x,5 + av) are linearly independent (mod g+(x,5 + av)), where g+(x,q ) = g+(x,q  OiN).
+
D & U I ~ ;= \ ~O; ( j = 0,. . . , m  I),
(k = 1, . . .,p)
+
+
Conversely, we have
imply that u = 0 in X'.
Theorem 9. Let the operator P be x,hyperbolic and the matrix p(x, 0, 5') be smoothly diagonalizable in X . Suppose that S is noncharacteristic, p = M + , and that on S the condition (2.10) and the uniform Lopatinskij conditions (a) and b)) hold. Then the mixed problem (2.1)(2.3) is strongly L,well posed and, moreover, the conclusion of Theorem 7 regarding the uniqueness holds.
In conclusion, we note that we could have required that the estimate Ilu; X t l l m  l + s
I'
for all to, t E [T, T,], to < t, y > 0, u E C" and s E Z+, with constants independent of u, to, t or y.
Theorem 7. Let the operator P be strictly x,hyperbolic in X , S be noncharacteristic, and let p = M'. Assume that the uniform Lopatinskij condition holds on S and that the boundaryvalue problem is normal. Then the mixed problem (2.1)(2.3) is strongly L,well posed. Furthermore, let X: 2 X , a x ' = X ; v XT+v S' v S , where S c S, S' E C". I f at each point x E S' the outward normal v'(x) to S' lies in T(g,N), then the facts that u E H m ( X ) ,Pu = 0 in X' and
Bk~Is,\s;= 0
+
+ Ilu; S(to,t)llml+s
The proof of this theorem rests on deducing the energy estimate by the symmetrizer method (Kreiss [19701) and then applying functionalanalytic arguments. Note that the symmetrizer method is much more complicated for the mixed problem than it is for the Cauchy problem. We also note that the problem for strictly hyperbolic equation, examined in the preceding section, can be reduced, after localization, to a problem for a pseudodifferential system of first order.
(2.28,)
instead of (2.25,), holds for all to, t E [T, T,], to c t, u E C" and s E Z+with a constant that is independent of to, t or u. Using the Duhamel's principle, we can easily show that the estimates (2.25,) and (2.28,) are equivalent; moreover, (2.28,) implies (2.28,) for an arbitrary s E Z+. Using (2.28,), we find that the smoothness condition on the righthand side can be relaxed:
Dif E Ll([T, T,], H"'(o)) ( j = 0, ..., s).
' '
1.4. Strong L,wellposedness of the Mixed Problem for Systems of First Order. In a series of works (Friedrichs [19581, Lax and Phillips [19601)devoted to the symmetric systems with "nonnegative" boundary conditions, there appeared L,estimates in two forms leading to the notions of strong and weak
1.5. Necessary Conditions for C"wellposedness of the Mixed Problem. Using the methods of $ 2 of Ivrii and Petkov [1974], we can easily prove the following
Theorem 10. Let X E S:, p = M+(X)and 1(X, 5 + r N ) = 0 for some 5 E lR"" and r E C\lR. Assume that l(X, 5 + ' N ) f 0 on C  . Then the mixed problem (2.1)(2.3) is not C"well posed.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
Corollary 1. Let X E S:, p = M+(X) and 1(X, N ) # 0. Then for the problem (2.1)(2.3) to be C"well posed, it is necessary that the Lopatinskij condition (2.26) holds with x = X.
the wave equation with Neumann boundary conditions. On the other hand, the wave equation with homogeneous Neumann condition possesses the classical energy integral and for it the existence and uniqueness theorems hold. Similar is the case with many problems for the Maxwell system for which there are no strongly L,wellposed boundaryvalue problems at all because the boundary is characteristic. In view of this situation, in their works Agemi and Shirota C1970, 19711introduced various other definitions for L,wellposedness.
200
Theorem 11. Let X E S,' and p = M+(X).Assume that condition a) is saitsfied for x = X, 5 = 4 and z = 7 E C\lR. Then for the C"wellposedness of the mixed problem (2.1')(2.3') it is necessary that the Lopatinskij condition holds for x = X, = and for all z E C \ l R .
Definition 4. The mixed problem (2.1)(2.3) is said to be weakly' L,well posed if it is C"well posed and if for all to, t E [T, T+], to < t, y 2 0 and u E C" such that Bkuls = 0 (k = 1, ... , p), s E z+,the a priori estimate
Corollary 2. Let X E S$ and p = M+(X).Assume that condition a) is satisfied for x = X and 5 = N . Then for (2.1')(2.3') to be C"well posed, it is necessary that condition a) is satisfied for x = X and all ( E R"+'and z E cC\lR.
J"J tt U ;x(to,r)It
mI+s,y
Definition 3. Let X E S,'\S, and r be an open convex cone with vertex X; r\X c {xo < KO}. We say that the problem (2.1)(2.3) satisfies the uniqueness condition ( U r )if the facts that P u = 0 in X,' nr and D&UIxTnj
=0
BkUIS+Tnr=O
( k = I, ...,/A)
imply that u = 0 in X : n r. Theorem 12. Let X E S,'\S, l ( X , .)
and p
+0
= M+(X).Suppose
that
on lR"+'  iT(g, N).
(2.29)
holds with the constant independent of to, t, y or u.
As compared to (2.25,), the member Ilu; S(to,t)llml+s,y on the lefthand side is now absent and a condition has been imposed on u that makes the members, absent on the righthand side, vanish. We shall assume throughout that S is noncharacteristic and the boundaryvalue problem is normal. Then it is not difficult to prove that (2.32,) implies (2.32,) for all s and that the problem (2.1)(2.3) has a solution
For the problem (2.1)(2.3) to be C"well posed and condition (Ur)to be satisfied, it is necessary that (2.30)
+ IIU; xt It m1 +s,y (2.32)
'
( j = 0, . .., m  l),
s+m1
E
n
j=O
C ~ C TT+], , ~ s + m  i  j (4)
for any
{ f, g j ( j = 0, .. . ,m  l), cp,(k = 1, . . .,p ) } E
and, in particular, that l ( X , N ) # 0.
(2.31)
ve E r(g,N): {(e,x  X) 2 o} nT = {x},
n{ q k E Hs+mmk1/2
~s+m3/2lal
(S,' 1
for la1 < s + m  2. Thus, in comparison to the strong L,wellposedness, the required smoothness of boundary data increases by 1/2, while that of DauIs decreases by 1/2. Moreover, the concept of L,wellposedness with the loss in smoothness by q E Z+was also introduced. Apart from the C"wellposedness, the estimate
and, in particular, for 8 = N . The present author is not aware whether the proofs of these results have been published in their general form somewhere. Corollary 1 of Theorem 10 has been proved by Wakabayashi [1980]. 1.6. Weak L,wellposedness of the Mixed Problem. The uniform Lopatinskij condition, introduced in 9 2.3 and 0 2.4, is an extremely severe restriction on the class of problems under discussion because it does not hold, for example, for
(S,+)}.
Then, by the embedding theorem, Dau
Theorem 13. Let X E S,'\S, and p = M+(X).Suppose that condition a) holds for x = X and for some 5 = 4 E lR"" and z = 7 E C \ R . For (2.1')(2.3') to be C"well posed and condition ( U r ) to be satisfied, it is necessary that condition b) holds for x = X,5 = 8
20 1
(2.33) must be satisfied for all u E C" having the property that Bkuls= 0 (k = 1, . . .,p) and u = 0 for xo < T. Even for q = 0, this definition is weaker than Definition 3, at least formally. When P and Bk are homogeneous operators with constant coefficients and o = lR"' x lR+,necessary and sufficient conditions for the Agemi and Shirota termed this problem, in contrast, as strongly L,well posed.
V.Ya. Ivrii
202
L,wellposedness with a loss in smoothness by q were given in terms of the Green's function of the problem (2.12)(2.13), and it was shown that for q = 0 the cone of dependence of the mixed problem coincides with the intersection of X: and the cone of dependence of the Cauchy problem (for q 2 1 this result is, in general, not true). the weak L,well posedness of Finally, it was shown that if w = R"'x E+, (2.1)(2.3) implies the weak L,wellposedness of the problem obtained by freezing the coefficients at any point X E S:, and, in particular, implies (2.19). This enables us to obtain conditions that are necessary for the weak L,wellposedness, which in many cases turn out to be sufficient as well. Let us examine in some detail the strictly hyperbolic operators of order m = 2. Let o = R"'x R+ and S be noncharacteristic and nonspatially similar. Then for x E S , 5'' E IR"' and 5, E C \ R , the polynomial p(x, to,5", 5.) has two roots t,, = A*(x, to,5") one with positive and another with negative imaginary parts. Then p = 1, and l(x, 5 0 , 5") = b(x, 5 0 , 5", A'(x,
50,
<"I),
where b is the principal symbol of B. We recall that a necessary condition for the problem (2.1)(2.3) to be C"well posed is that the Lopatinskij condition holds. Namely, l(x, to,5") # 0 for x E S:, 5" E R"'and 5, E C\R (we assume that (2.19) holds, that is, that l(x,  i, 0) # 0).A necessary and sufficient condition for the problem to be strongly L,well posed is the uniform Lopatinskij condition, namely, l(x, lo,5") # 0 for x E S:, 5" E R"'and 5, E CC. This condition is fulfilled, in particular, for the Dirichlet boundary condition with B = I; it is assumed that the boundary condition is normal.
Theorem 14 (Miyatake [1975]). Let
B=
2 bj(x)Dj+ b'(x),
then either c
5" E IRnl,
*
Then the problem (2.1)(2.3) is weakly L,well posed.
We note that if (2.35) is violated at some point X E S:, we can impose other technical conditions that guarantee weak L,wellposedness (see Miyatake [1975], where (2.34) is given in various other forms). It turns out that if conditions (2.34) and (2.35) or some of the abovementioned technical conditions are satisfied, then the conoid of dependence of the mixed problem (2.1)(2.3) will be the intersection of X: and the conoid of dependence of the Cauchy problem. We also note that (2.34) implies that the Lopatinskij determinant 1 does not vanish at hyperbolic points of T*S:\O but may vanish at tangent points and at elliptic points (the classification of points of T*S\O is given in the next section). We have already discussed the question of the loss in smoothness in the case of weak L,wellposedness. But in some cases the actual loss is smaller. Let P = 0 be the wave operator and X = IR x o.Consider the maps: supp v c s,++ u E C"(X): Pu = 0,
u E C"(S),
If w is strictly concave (that is, if o is an "exterior" domain with a nonzero Gaussian curvature of boundary), then A is an Airy type pseudodifferential operator of class L&3 (Melrose [1978]). This is a classical pseudodifferential operator outside the tangent set 9 and the inverse operator A' is an Airy type What is more, A' is a classical pseudodifferential operator of class L&,. pseudodifferential operator of order ( 1) outside 59. Therefore for a solution of the problem
Pu=O,
where b,, = 1. (i) I f P and B have constant coefficients, then a necessary and sufficient condition for the problem (2.1)(2.3) to be weakly L,well posed is the inequality
E ST+,
= 0 or Im b, = . . = Im bn' = 0 in some neighbourhood of X, where c = Re b,  PEnE, . P;in.
j=O
vx
203
11. Linear Hyperbolic Equations
5,
E
c
(2.341
(of course, in this case 1 and A+ do not depend on x). (ii) I f P and B have variable coefficients, then (2.34) is a necessary condition for the problem to be weakly L,well posed. (iii) Assume that the coefficients of P and B are constant for ( X I ( > R and that (2.34) as well as the following additional condition hold: rf X E s;, c(X) = 0, (2.35)
a"l an s = w
suppucx;,
the estimate
[Iu;
x;
llkfl
t
st
Ilk+'
cllw;
st
llk+l/3
holds for all k E Z?. This estimate cannot be improved upon. 1.7. C"well Posed Mixed Problem for Strictly Hyperbolic Equations of Second Order. Let us begin with the results of Ikawa and Soga for the wave equation. Let o c IR"", ao be compact and v the inward unit normal to do.Let P = 0 and B = &, bjDj + b', where bjvj = 1 and bj are real. A necessary and sufficient condition for strong (weak) L,wellposedness is the inequality b, P < 0 (b, + fi < 0), where = bj'  1)'I2. By Theorem 10, a necessary condition for the C"wellposedness is that b, < 1 for n 2 3 and b, < (1 + fiZ)'" for n = 2.
+
(cy=l
204
205
11. Linear Hyperbolic Equations
V.Ya. Ivrii
A domain X is said to be bicharacteristically strictly convex (concave)if
Theorem 15 (Ikawa [198l]). Let n = 3. Let w be a strictly concave domain. For the mixed problem (2.1)(2.3) to be C"well posed and to have a finite propagation speed, it is necessary and sufficient that bo < 1.
X,
that is, if
Now let n = 2. Let bo = 0, bj be independent of xo, z = z(s) = blv,  bZvl, A = A(s) = blvl + bzvz and x r = xr(s),where s is a parameter on am. We drop the earlier assumption that 1E 1.
"41
= 0,
{ p , x,} = 0
H:x,
>< 0,
2 0. The behaviour of bicharacteristics in bicharacteristically
ax,
convex and concave domains is depicted in Fig. 2 and Fig. 3, respectively.
+
Theorem 16 (Ikawa [198l]). Let 1 1. Let o be a strictly concave domain. For the problem (2.1)(2.3) to be C"well posed, it is necessary and suficient that z and dz/dS do not both vanish simultaneously.
xnf
Theorem 17 (Soga [1980]). For the problem (2.1)(2.3) to be C"well posed, it is necessary that at no point of aw does 1(s) change its sign. Theorem 18 (Soga [1980]). Assume that z(s) does not vanish and I(s) does not change its sign on aw. If [1(~)]'/~ is a smooth function, then the mixed problem (2.1)(2.3) is C"well posed but does not have, in general, a finite propagation speed.
x
We now turn our attention to far more general results of Eskin. Let P be a strictly x,hyperbolic operator of second order and B a differential operator of an arbitrary order M . We examine the mixed problem in a somewhat different form: Pu = 0, supp u c x;, Buls = cp. (2.36)
S Fig. 2
Theorem 19 (Eskin [1979180, 19811). If the domain X is bicharacteristically strictly concave and i f conditions (2.26), (2.38) and (2.39) hold, then the mixed problem is C"well posed. Furthermore, if cp E Hfo;M"(S) and cp = 0 for xo < T, then u E H",,(X).
Suppose that S is noncharacteristic and nonspatially similar. Then locally, in a suitable coordinate system,
X
= {xn 2 O},
P(X,
5 ) = 5.'  q(x, 5 0 , t"),
Theorem 20 (Eskin [1979/80]). Let the domain X be bicharacteristically strictly convex. Let conditions (2.26'), (2.38) and (2.39) be satisfied. Assume that in a neighbourhood of any point p E 9 n { I = 0} the condition
where q = 0 * qto # 0.
(2.37)
Let T*S\O 3 %? = {q > 0},6 = {q < 0 } and 9 = {q = 0} be, respectively, a hyperbolic, elliptic and tangent set (or set of hyperbolic, elliptic and tangent points, respectively). Let l(x, 5 0 , 5") = b(x, 50, t",A+(%
50,
where Im
l(x, to,5") # 0 for Im to< 0
VP = ( x , 5 0 9 5 ' 7 E 9 n { I t o l
(2.26) i
50,
t")= 0,
(x, to,(?)
E2
u8
l(x, 5 0 , 5") = 0, ( x , t o , 5") E 9 =>
ai (x,
50,t") # 0,
(2.38)
a50
ab ag. (x, t o , t")# 0.
(2.39)
+ 157 = 1}
(2.40)
is fulfilled; here 5, = y(p) is a smooth root of the polynomial b(p, 5,) such that y ( p ) = 0. Then the mixed problem (2.36) is C"well posed. Furhermore, if cp E HfOYM"(S), cp = 0 for xo < T and 0 < t  T is sufficiently small, then u E Hfoc(x;).
2 0 for
holds and that
I(x,
IRe Y(P)l InllRe Y(P)l sign 4ro(P)G CIIm Y(P)I2
57)
be the Lopatinskij determinant and ' 1 = &,/ Im to< 0. Assume that the Lopatinskij condition
Fig. 3
Theorem 21 (Eskin [19791803). Let the domain X be bicharacteristically strictly convex. Let conditions (2.26), (2.38) and (2.39) be satisfied. Assume that { y , q} (p) # 0 at any point p E 9 n (1 = 0 } in a neighbourhood of which (2.40) is not satisfied. Then the mixed problem (2.36) is C"well posed. Furthermore, if cp E H/o;M+L+l(S),cp = 0 for xo < T and 0 < t  T is sufficiently small, then u E Hfo,(X;), where
V.Ya. Ivrii
206
and the maximum is taken over all the points of which the condition (2.40)fails to hold.
11. Linear Hyperbolic Equations i j E 9 n( 1
These assumptions ensure strong L,wellposedness for the mixed problem in the quadrant ( x o > 0, x , > O } that is obtained from the original problem by removing all differentiation operators other than Do and 0,and the variables x l , . . .,x ,  ~are regarded as parameters. Under these assumptions, we can construct Pcharacteristic surfaces Ak that contain So and are orthogonal to the vectors (&(x),0, . ..,0, 1) at points of So. When k 6 M + , these surfaces lie in X i for xo 2 0 and divide X i into M + 1 sectors T,(k = 0, 1, . . .,M + ) (see Fig. 4).
= 0 } in a neighbourhood
On the other hand, we have
Theorem 22 (Eskin [1979/80]). Let the domain X be bicharacteristically strictly convex. Assume that there exists a sequence of points p k € 9 n { I toI I ["I = 1>, p k + p, such that fqc,,(p) > 0, fRe Y ( p k ) < 0,
+
(Imr(pk))2 In
IRe
y(pk)l
(Re Y ( p k ) )  ' *
as
+
201
+
O0
and
S
I {Y, 4 ) ( p k ) l
< CIRe
r(pk)ld,
> O.
Then the mixed problem (2.36) is not C"well posed. We conclude the present section with the remark that for nonstrictly hyperbolic equations lower terms also figure in the wellposedness conditions. Thus, if x = IR' x E+, P = D,Z  Dl  x ~ D :+ cD1,
c E (c,
(2.41) k
and B = D, + aD,, then the mixed problem is Cwell posed for Re a < 0 and for Re a = 0, c E IR; in the remaining cases it is not well posed. When B = I , a necessary and sufficient condition for the wellposedness is that c E IR (Zajtseva [1983]). Similar, though more complicated, wellposedness conditions have been obtained by Zajtseva [1987] for the operator
+ cD1,
x, Fig. 4
(2.42)
i
1.8. Mixed Problem in the Classes of Analytic Functions. We formulate Duff's theorem 119593, which is an analogue of the CauchyKovalevskaya theorem for the Cauchy problem. Let o = ( x , > 0 } and T = 0. Let P and Bk (k = 1, .. .,p) be scalar differential operators of order m and mk, respectively, whose coefficients are analytic in a neighbourhood of 0. Assume that
1
P
= 0:  Dl  x$D,
n
c E C.
1
1
m
P(X, t o ,
where Aj(x) are real, Aj # M + + 1. We also assume that
jlk
0, t n ) =
j=1
( t o  Aj(x)tn),
(2.43)
for j # k, Aj c 0 for j iM + , and Aj > 0 for j
>
Theorem 23. Let conditions (2.43) and (2.44) be satisfied. Let the functions f, (pk (k = 1, . . . , p) be analytic in a neighbourhood of 0. Then in the neighbourhood of 0 there exists a function u E L, satisfying (2.1) in X i in the sense of the theory of generalized functions and the boundary conditions (2.2) on S,'\So and the initial conditions (2.3) on Xo\So. Moreover, its restrictions to can be continued to functions that are analytic in a neighbourhood of Further, iff, gj and (pk satisfy the compatibility conditions of order s, then 0. u E Cs+m2and the solution is unique. NoG that the compatibility conditions of infinite order enable us to determine the analytic functions (pk uniquely, and in this case the solution u can be obtained by the CauchyKovalevskaya theorem. We also note that if the mixed problem (2.1)(2.3) is C"well posed (in general, the coefficients of P and Bk are no longer analytic), conditions (2.43) and (2.44) hold, and the functions f, g j and (pk belong to C", then in a neighbourhood of So the solution is infinitely smooth in each of the sectors up to its boundary and belongs to Cs+mn2 iff, g, and (pk satisfy the compatibility conditions of order s. gj ( j = 0, . . .,m  1) and
1 1
E
p = M', the boundaryvalue problem is normal, and the polynomials
bk(x,to,0, 5.) are linearly independent (mod p+(x,to,[,,)),
I
where (2.44)
0 2.
Propagation of C"Singularities
In this section we examine the propagation of Cmwavefronts of solutions to the boundaryvalue problems for hyperbolic equations and systems. However, we shall not discuss the wellposedness question for these problems.
208
V.Ya. Ivrii
11. Linear Hyperbolic Equations
2.1. The Wave Fronts. Let X = X x IR = IR" x R' 3 (xo, ..., x,~, x,), S = X ' x 0 and 8 = X\S. Let 1 : T*XIs f T*S be the natural map. We denote by Lm(X')the space of classical pseudodifferential operators of order m on X and set L""(X) = Cm(IR+,Lm(X')).We shall discuss only operators with compact supports. We introduce the function space
where (., .>+ denotes the scalar product in L2(lR+,ad), the Hermitian product in (EL and yu = (u, D,u, ...,On m1 U ) I ~ , = ~ are the Cauchy data of u on S, D = md, and T ( p )is a D x D smooth Hermitian matrix, with rank T(p*)= D, that can be expressed in an obvious manner in terms of ak(p). Let D , and d , denote, respectively, the number of positive eigenvalues of the matrices T ( p * ) and ao(p*). Then D, = md/2 for even m and D, = (m  l)d/2 + d , for odd m. Let B = (Bo,. . . ,Bml)be the boundary operator and Bk E L&,,(s). Let bk, b = (bo,. . ., bml),the principal symbols of Bk and B, respectively, be D, x d and D, x D matrices. We assume that ( a ,
%"(X) = {u, D i u E L2(E+,H S  ' ( X ) ) Vj E Z '}.
Definition 5. By the boundary wave front (boundary wave front of order s) of we mean that set WFb(u)c T*S\O (WFi(u) c T*S\O): a distribution u E 9'(8) p $ WFb(u) ( p # WFi(u)) if we can find an operator a(x, D')E Lo'(X) which is elliptic at the point p and is such that au E C m ( X )(au E %'(X)). Unfortunately, the invariance of such a definition under a change of the variable x, is very doubtful. However, in many cases this invariance can be established for solutions of systems with smooth righthand sides. Another definition of the boundary wave front can be found in Melrose [1981]. We also introduce the complete wave fronts
rank b(p*) = D+,
 Tb) + P(P)b(P)+ b*(p)P*(p) is a nonnegative definite matrix in the neighbourhood of p*. We already know that for u E 9'(8)
= WF'(U)u z  ~WF~(U).
Thus "gluing" of T * X on S appears already in the definition: two points 8 and 8' E T*XIs are identified if ztl = 10'. 2.2. Propagation of Singularities of Solutions to Dissipative BoundaryValue Problems for Symmetric Hyperbolic Systems. We consider a d x d matrix operator m P= AkD,"k, (2.45)
1
k=O
, t1 1
where A , E L$(X). Here the notation " d l x d2" stands for the size, and "d x d" will be written as "d". Let ak and
1
m
p=
1
akt,"k
k=O
be the principal symbols of A , and P, respectively. Let T*S\O 3 p* be the point in a neighbourhood of which the discussions will be carried out. We assume that
(that is, S is noncharacteristic at p*), and that
f
P* = P
(2.47)
(that is, the system is principally symmetric). Then for all u, v E Y(IR+, ad)and all p E T*S\O lying in a neighbourhood of p*, we have ( P h D,)u, v>+  (4P ( P , D,)o>+
I
(2.46)
rank ao(p*) = d
=
i ( T ( p W , YO),,
(2.48)
( T ( p ) U , U>D < 0 VU E Ker b(p) (2.49) in some neighbourhood of p*, that is, there the boundary operator B is nondegenerate and is dissipative. Then there exists a D x D, smooth matrix P ( p ) such that
WFf(u) = WF(u) u ZlWFb(U), WF;
209
WFs(u) c WF""(Pu) u Z, where C = Char P = (9 E T * X , g = det p = 0} is the characteristic set of operator P ; outside this set P is elliptic. Let E ( p ) denote the solution space of the system ( p , DJu = 0 on ,'RI the solutions decaying as x, +a. Definition 6. The problem ( P , B), satisfying conditions (2.46) and (2.47), is elliptic at a point p E T*S\O if p 4 zZ and the map b(p)y:E ( p ) + Cpis a bijection. Here p = dim E ; p = D, if (2.47) holds and p I$ zZ. If Cbdenotes the set of points where the problem (P, B) is nonelliptic, Zb is known as a characteristic set of the problem (P, B). By the standard methods of the theory of elliptic boundaryvalue problems, we can easily show that if (2.46) and (2.48) hold and u E C m ( R + ,9'(X)), then c WF:'"(pu) U WF"i'2(Byu) U Cb. (2.50) WF~(U) = zk ( k = 1, . . . , N ) be all the real and distinct roots of the Finally, let characteristic equation g(p*, t,) = 0. Theorem 24 (Ivrii [1986]). Assume that conditions (2.46), (2.47), (2.48) and (2.49) hold. Let (pk E C m ( E + x (T*X'\O)), cp E Cm(T*S\O) be red, positive homogeneous functions in (' of degree 0 such that cpl = * * * = cpN = cp on T*S, cp(p*) = 0 and ({P?V k } ( p * , z k ) v ? v> (2.51) k = 1, .. . , N , v v E Ker p(p*, zk)\o,
<,
'
Re({p, q } ( p * ,
Dn)u,
O>+

Re i(B(p*){b,(P}(p*)Yv?y v > D
V v E E ( p * ) n Ker b(p*)y\O.
'
(2.52)
V.Ya. Ivrii
210
If
11. Linear Hyperbolic Equations
Remark 3. Assume that (2.46)(2.48) hold. Let dim Ker p(p*, zk) = 1 for all k = 1, .. . , N . Let rk denote the multiplicity of zk, a root of the characteristic equation. Assume that Ek is the space of generalizedexponential solutions on ]R+ of the system p ( p * , D,)v = 0 that correspond to the characteristic root zk and to the following growth condition as x,, + +a:
u E Cm(R+,9 ' ( X ' ) )and
p*
4 WFbSm+l(Pu) u WF"+"'(Byu),
(2.53)
WF"(u)n{cpk < 0 ) n wk= @ Vk = 1, ..., N ,
(2.54)
WFi(u) n (cp < 0 } n W = (ZI,
(2.55)
211
u = 0(x2"")
where w k and W are sufficiently small neighbourhoods of the points (p*, zk) and p* in T * X and T*S, respectively, then p* 4 WFi(u). The same assertions remain valid for the wave fronts of infinite order also.
for even rk, (2.58)
u = O ( X ~  ~ ) / ~for ) odd r,
and for
In analogy with the established terminology of 5 3.1, it is natural to call P, satisfying (2.47) and (2.51) with a given k, a microhyperbolic operator at the point ( p * , zk) in the direction Hqk(p*) and the problem (P, B), satisfying (2.46)(2.49), (2.51) and (2.52),microhyperbolic at the point p* in the multidirection
while u = O ( X ~  " / ~ ) for odd rk and ak > 0,
(2.58")
where (pk satisfies (2.51). We set = El 0 E , 0 ... 0 EN 0 E(p*). If the strict dissipativity condition is satisfied, then Remark 1. If the nondegeneracy condition E(p*) n Ker b(p*)y = 0,
n Ker b(p*)y = 0.
(2.56)
Conversely, if (2.46)(2.48) hold and
is satisfied, then (2.52) of Theorem 24 is empty and then the propagation of singularities does not depend on any concrete choice of the boundary operator B (though for some B there may be abnormally poor propagation of singularities). This theorem has been proved roughly under such an assumption by Ivrii [1979d, e, 1980al. If p* 4 zC,then the study of the propagation of singularities of solutions to boundaryvalue problems reduces to that of propagation of singularities for a certain operator W E L!,2(S).Its characteristic set is Z,,. The Rayleigh waves are a phenomenon of this very type in which W is a symmetric operator having characteristics of constant multiplicity 2. In the general case, there is a complicated combination of the singularity propagation. A detailed examination in the particular case of the wave equation can be found in 5 2.6.
i 1
I
Remark 2. If the condition of strict dissipativity
(2.57) holds at the point p*, then (2.56) also holds. What is more, in this case lower smoothness is required from the boundary data if (2.53) is replaced by p*
# WF~'"+'(Pu)u WFs(Byu),
yielding, in addition, that p* $
u
k=O
(2.53')
1
1 k
!
m
WF"k(D,k~I,).
(2.59)
dim Ker p ( p * , z k ) = 1 for all k
=
1, ..., N ,
then the problem satisfying (2.59) can be reduced to a strictly dissipative problem for a symmetric system of first order.
2.3. The Geometry of the Propagation of Singularities. Assuming that the nondegeneracy condition (2.48) holds, we list a number of concrete results. We start with the assumption that P has simple characteristics. Analogous results also hold for operators with characteristics of constant multiplicity, but in this case g is not the characteristic polynomial of P but a polynomial of principal type whose zeros coincide with those of the characteristic polynomial. We shall assume throughout, without stating it each time, that the righthand side and the boundary data are infinitely smooth in the desired domain I/ c T*X\O of the phase space, that is, it does not intersect WFf(Pu) or z~WF(BUI,). We immediately find that if the only bicharacteristics are those transversal to S , then the wave fronts of the solutions arrive on S along the incoming bicharacteristics and move away from S along the outgoing ones. In this way they are propagated along bicharacteristic billiards having branches, in general, on S. We can easily drop the transversal bicharacteristics from the microlocal analysis, which indeed we shall be doing for the convenience in formulation, without stating it each time. Suppose that over the point p* E T*S\O there lies exactly one tangent bicharacteristic, that is, the characteristic polynomial g ( p , t,,)has for p = p* exactly one multiple real root 5, = zl. Let 8* = ( p * , zl). We recall that since the characteristics of P are simple, it follows that g,(8*) # 0. We also assume that
V.Ya. Ivrik
11. Linear Hyperbolic Equations
z1 is a root of multiplicity 2 of the characteristic polynomial, that is, g(O*) = (Hxng)(fl*) = 0, Hing(O*)# 0. To be definite, we assume that H:ng(O*) > 0. If the domain X is bicharacteristically strictly convex at 0*, that is, if (H,2xn)(O*)< 0, then in the neighbourhood of zlp* the wave fronts are propagated along bicharacteristic billiards of g as well as along boundary bicharacteristics, namely, along bicharacteristics, defined on T*S\O, of the boundary symbol gb(p) = g ( p , y(p)), where 5, = y ( p ) is a root of the polynomial { g , x,} { p , t,} such that y ( p * ) = zl.A boundary bicharacteristic is the zprojection of a curve, lying in T*XI,, along which
of singularities in bicharacteristically concave domains near the points of tangency. This is due to the requirement that the function rp should not change when the bicharacteristic is reflected from S , that is, that this function should not depend on 5,. True, some implicit and weak dependence was assumed for N 2 2 because we had considered a collection of functions rpk. We now drop this requirement and impose a more natural condition that rp does not decrease when the bicharacteristic is reflected from S. However, we are forced to examine a considerably narrow class of systems and only strictly dissipative boundaryvalue problems for them. We assume that the following conditions are satisfied. a) In a neighbourhood of L'p* in T*Xls, the characteristic symbol can be factorized
212
dP  = Hg(P) + XHXn(P), dt
(2.60)
N
g = ( 9 , xn} = 0;
this curve will also be referred to as a boundary bicharacteristic. Note that x = (Hixn)(H:ng)l.A boundary bicharacteristic will be termed a C"boundary bicharacteristic if H i x , < 0 on it. Such bicharacteristics are limits of those bicharacteristic billiards which press against the boundary more and more tightly. In particular, for the wave equation bicharacteristic billiards are the usual billiard trajectories, while boundary bicharacteristics are the geodesic boundaries. If X is bicharacteristically concave, that is, if Hfx, 2 0, then C"singularities are, perhaps, not propagated along boundary bicharacteristics (see the next two sections). Therefore, in this case Theorem 24 does not furnish a precise description of the propagation of C"singularities. If there are several tangent bicharacteristics over p * , each satisfying the assumptions made above, and if X is a bicharacteristically strictly convex domain relative to each of them, then there arise several boundary symbols g b , & and there may be different types of propagation depending on their tion". The simplest case is the one where all the boundary symbols g b , &
~~~~~~~~~
( k = 1,. . ., N) are in involution and the vector fields HBb.l, ..., Hgb." ('7
U
are 85 linearly independent. Then there arises the propagation of singularities along Ndimensional bicharacteristic sheets that are constructed in accordance with the aggregate of boundary symbols in exactly the same way as was done in 0 3.2, Chap. 1, in accordance with the aggregate of factors. Ivrii [1979d, e, 1980al has also analysed the propagation of singularities for the case where one tangent bicharacteristic lies over p* and the root 5, = z1 of the characteristic polynomial is of multiplicity 3. The propagation of singularities of solutions to boundaryvalue problems for systems with characteristics of variable multiplicity has been studied to a lesser extent, and for concrete results the reader is referred to Ivrii [1979d, e, 1980al. 2.4. The Propagation of Singularities of Solutions to Strictly Dissipative BoundaryValue Problems for Symmetric Hyperbolic Systems. Theorem 24, though fairly general, does not enable us to study conclusively the propagation
213
g =e
n
k=l
(2.61)
h?,
where e is an elliptic symbol and hk are real polynomials in 5, of degree mk = 1, 2 with the stipulation that hj and h,, j # k, do not both vanish simultaneously and that hk = 0 * {hk,xo} > 0. b) In a neighbourhood of p* there exists a real, positive homogeneous symbol q of degree 0 such that q ( p * ) = 0, (q, xo} ( p * ) # 0 and for q ( p ) 3 0 all the roots 5, of hk(p,5,) are real, whereas for q ( p ) < 0, mk = 2, all the roots of hk(p,5,) are nonreal. Then both the roots of h,(p, 5,) coincide if q ( p ) = 0 and mk = 2. Condition b) clearly rules out the presence of bicharacteristic boundary sheets of dimensuion greater than 1. Theorem 25 (Ivrii [1980b]). Suppose that conditions a) and b), conditions b) of Theorem 37, Chap. 1, and conditions (2.46), (2.48) and (2.49) hold in a neighbourhood of z  l p * . Let rp E C"(T*X\O) be a real positive homogeneous function of degree 0 that satisfies (1.38) (of Theorem 37, Chap. 1) and the conditions hk(O')= hj(O")= 0, {hk, x n } ( e ' ) 2 0 2 {hj, xn)(e), let
=
q=O
in the neighbourhood of
zlp*.
p*
z e rp(e')2 rp(e);
(2.63)
on z  l p * n { g = o } If u E Cm(]R+,W ( X ' ) )and
# WFi(Pu) u WF"(Byu),
WF;(u)n { g = 0} n {rp < 0} n W =
0,
where W is a neighbourhood of zlp* in T * X , then p*
(2.62)
4 WFE(u) u
u WF"k(D,k~I,). m
k=O
(2.64)
215
V.Ya. Ivrii
11. Linear Hyperbolic Equations
We do not know whether this theorem remains valid for wave fronts of infinite order. However, applying it to concrete cases, we obtain results for wave fronts of finite as well as infinite order. For N = 1, we can easily obtain a global version of this theorem that is also true for infiniteorder wave fronts. Theorem 25 is applicable to strictly hyperbolic operators of second order. Namely, to operators of the form
If E C and z'zV c V is a neighbourhood of we denote by K'(e, V) the union of all Cmraysthat issue from 8 in the direction of increasing (decreasing) xo and lie in It can be shown that K'(8, V ) have the properties (1.43)(1.46). Further, if the domain X is bicharacteristically concave or its boundary S is bicharacteristically nondegenerate, that is, if the set
214
P
=
D:
+ Q,
Q E L2'(X),
e
e,
Qm = { p E 3, ( H , k ~ , ) ( p=) 0
Vk E Z+}
(2.65)
that are such that a) q, the principal symbol of Q, is real and q 2 0 =. E { q , xo} > 0, E = & 1 fixed, if the boundary operator B = yo& y', y j E Lj(S), satisfies (2.59). This condition in this case is equivalent to the two uniform Lopatinskijtype conditions
+
(2.66)
is empty or, more generally, each point p E 9" is an interior point of some bicharacteristic of g lying in Q (in particular, if g is an analytic function), then through each point 8 E C there passes a unique maximally extended C"ray y(@. On this ray the exceptional points (the points where the bicharacteritic is reflected from S and the points where transition takes place from the bicharacteristic to the C"boundary bicharacteristic and viceversa) constitute a locally finite set. In the general case, however, CODrayshave branching on Qm, and together with them the wave fronts too have branching. Theorem 26. Let the abovementioned assumptions be fulfilled. Let z'zV c V be an open set, V n Z C C and E C n V. I f
IZ > 0 * b(8 + AiHJ # 0,
\
I
(WF,(Pu) u z'WF(Bul,)) n K  ( & V) = @,
where p and b are principal symbols of P and B, respectively. As we have already noted, such problems lead to strictly dissipative boundaryvalue problems for firstorder systems satisfying the conditions (2.46)(2.48) and condition b) of Theorem 37, Chap. 1. When y o = 1 and y' E Lo(S),such a reduction is not possible in the neighbourhood of points of tangency. However, in this case Melrose and Sjostrand [1978,1982] have obtained similar results, as we describe below. 2.5. The Geometry of the Propagation of Singularities (the Concluding Part). We assume in this section that the operators P and B satisfy the hypotheses of Theorem 25. We start by describing the propagation of singularities when the characteristic polynomial is of the form g = 5: + q, where the symbol q = q(x, <') satisfies condition a) of Theorem 25. Under these assumptions WF,(u) c WF,(Pu) n WF(Bu(,)u % u 3,
e
(2.67)
(2.68)
where, we recall, 2,8,and Q are, respectively, hyperbolic, elliptic and tangent sets. We shall consider the restriction of complete sets. We shall examine the restriction of the complete wave fronts to the manifold C = { g = 0}, identifying the equivalent points of Zls. Definition 7. By a C"ray we mean a Lipschitz curve y = (0 = e ( t ) } , with parameter t = xo,lying in C that in the neighbourhood of almost all its points is either a bicharacteristic or a C"boundary bicharacteristic.
WF,(u)n
K ( 8 , V ) n d V = 0,
(2.69) (2.70)
then
WF,(u)nK(8, V ) =
a.
(2.71)
We now assume that the hypotheses of Theorem 25 hold and that the decomposition (2.61) holds with N 2 2 in a neighbourhood of i'p* in T * X . Assume that the domain X is bicharacteristically strictly concave with respect to h, ( k = 1, . .. , M ) and M = N  1 or else M < N  1 and X is bicharacteristically strictly convex with respect to h, ( k = M 1, .. .,N ) . In the first case, the singularities are propagated along billiards that consist of bicharacteristics h, ( k = 1, . .., M  1) and Cooraysh,. In the second case, singularities are propagated along billiards consisting of bicharacteristics h, ( k = 1, . . ., N ) and boundary bicharacteristics, that is, bicharacteristics of thC boundary symbol. For M = 0 Theorem 24 can also be applied to the present situation. Therefore in this case (2.57) can be replaced by (2.56). We observe that for bicharacteristically strictly concave domains there is, in addition to the implicit study of propagation of singularities, an explicit construction of a parametrix (Eskin [1976], Melrose [1975], Taylor [1976a, b, 19791) that is based on Ludwig's constructions and uses Airy functions as the components of oscillatory integrals. In bicharacteristically strictly convex domains there is "semiexplicit" construction (Eskin [19771). There are also several works devoted to the reflection and refraction problems that fit into the given scheme.
+
216
11. Linear Hyperbolic Equations
V.Ya. Ivrii
2.6. The Propagation of Singularities of Solutions to Nonclassical Problems. In this section we shall examine the propagation of singularities of solutions to the nonclassical problems. That is, to these boundaryvalue problems for strictly hyperbolic equations of second order that do not satisfy the conditions (2.66) and (2.67). Let P be an operator of the form (2.65), B = 0, M , M E L'(S) and p its principal symbol. Then the sets of points where (2.66) and (2.67) are violated + p = 0} c 2 u B and A+\A', where are, respectively, the sets A  = {(E&
+
A+ = { i f i + p = O} c 6 u 4 A' = { q = p = O}
=A
nQ =2 n B
(we identify the points of T*XsIs with their Iprojections). Evidently, A U A + c A = (4 = p2}. If p E 6 and p 4 WFb(Pu),we can easily show that p 4 W F ( ( D ,  a)uls),where L'(S) 3 a is an operator with principal symbol a, = i f i in a neighbourhood of p. If, in addition, p 4 WF(BuI,), then p 4 WF(au), where ZI = uIs and a = i(a + M ) E L'(S) is an operator with principal symbol + ip and with characteristic manifold A+\A'. It is also easy to show that if p E 2,p 4 WFb(Pu)and K  ( p , V )n W F ( u )= 4, where V is a sufficiently small nighbourhood of the set zlpnZ, then p 4 W F ( ( D ,  fl)uls),where L'(S) 3 fl is an operator with principal symbol fl, =  E& in the neighbourhood of p. If, in addition, p 4 WF(Bul,),then p 4 WF(bu), where b = fl M is an operator with principal symbol E& + p and characteristic manifold A\A'. Thus outside the tangent set 92 the study of the propagation of singularities of solutions to boundaryvalue problems reduces to that of solutions to pseudodifferential equations on S. On the other hand, B satisfies (2.66) and (2.67) outside A + u A  . Therefore our main interest must lie in the propagation of singularities only near the set A'. It is evident that there a and b are operators of principal type.
6
+
Definition 8. By a generalized Cmray we mean a Lipschitz curve y = (6' = 8 ( t ) } c Z u A + u A  , with parameter t = x,,,that in the neighbourhood of almost all its points 8 E Z u A\Ao is either a C"ray or a bicharacteristic of b and in the neighbourhood of each of its points 6' E A+\Ao it is a bicharacteristic do of a, and, finally, if at each of its points 8 E A' it satisfies the equation  = dt ?fqQ r z 0. Let E V n (Z u A') and T*X\O 3 V be an open set such that z'zV c K We now denote by K*(g, V ) the union of all generalized Cmraysissuing from in the direction of increasing (decreasing) xo and lying in It can be shown that K'(8, V )have the properties (1.43)(1.46) if, at least, one of the following conditions is satisfied: o # O on A' (2.72) Rep=oImp,
217
(then the operators a and b are elliptic near A'); (4, Rep} > 0,
~ { qImp} , < 0 on A'
(2.73)
(then a and b are subelliptic operators near A'); Imp0,
{q,Rep} > O
on A'
(2.74)
(then the operator a is elliptic and b is an operator of real principal type near A', A is a C"manifold and A  is a Cmmanifoldwith edge A'; the bicharacteristics of b, parametrized by xo, start from A'); E R e p G O if q + ( I m p ) ' = O
(2.75)
(then the operator b is elliptic). The inclusion (2.67) remains valid if one of the conditions (2.72) to (2.74) holds. In the general case the inclusion WFb(U) c
wFb(PU) U WF(BU1,)U 2 u 9 u A +
(2.76)
holds. Theorem 27. Assume that (2.65) and one of the conditions (2.72)(2.75) hold. Let V be a sufficiently close neighbourhood of V n A' such that z'zV c V, 6' E V. Then (2.69) and (2.70) imply (2.71). Under assumptions (2.72)(2.74) this theorem has been established by Ivrii [1981] while under (2.75) by Melrose [1975]. Thus under assumptiions (2.72) and (2.73) the propagation of singularities remains classical, but under (2.74) there appears additional propagation of singularities along bicharacteristics of b and additional branching of singularities at points of A. If (2.75) is assumed to hold, there appears additional propagation of singularities along bicharacteristics of a and branching at points of A', though, possibly, not at all points. We note that under (2.75) a does not always have a real principal symbol a , . Therefore by a bicharacteristic in this case we mean the curve along which a = 0, where z( p ) is a nonvanishing complexvalued function.
4 3. The Propagation of Analytic Singularities We devote a separate section to the study of propagation of analytic singularities because in the case of the mixed problem it differs significantly from the propagation of C"singularities. At the same time, we shall use the notation of the preceding section. We basically follow the works of Sjostrand [1980a, b, c, 19811.
218
V.Ya. Ivrii
11. Linear Hyperbolic Equations
3.1. The General Theory. Let g , containing 0, be an open domain in IR"+'. Put X = 2 n (x, 2 0},S = 2 n {x, = 0}, and 8 = X\S. Let d ( X ) and d ( S ) denote, respectively, the space of functions that are realanalytic in X and on S. We introduce the analytic wave fronts
If V is a sufficiently small neighbourhood of p* in T*S\O and 0 < E is sufficiently small, and if p E V, t E (0, E ) and z is a root of the characteristic polynomial g ( p ito, z) that is close to zk, then Im(z  itCk)# 0 ( k = 1, . ..,M ) , by virtue of (2.31). Now suppose that R ' ( p , t ) is the set of roots z of this polynomial that are close to zk with some k = 1, . . ., M and with Im(z  itCk)> 0 and R"(p, t ) is the set of roots z of the same polynomial that are close to zk with some k = M + 1, . . ., N and with Im zk > 0. We denote by E ( p , t ) the space spanned over generalized exponential solutions of the system p ( p ito, 0,)v = 0 corresponding to R'(p, t ) u R"(p, t), the set of roots of the characteristic polynomial.
WF,(u) = WF,(uIj) c T*2?\O, wFb,(U)
=
u
WF,(D,kuI,) c T*S\O,
WF,,(u) = WF,(U) u l'wFb,(U). Let P be a d x d matrix differential operator of order m with coefficients in d ( X ) . Put B = (El, .. ., EN),where Bk are 1 x d matrix differential operators of order mk with coefficients in d ( S ) . Let p, bk and b be the principal symbol of P , Bk and B, respectively, and let g = det p be the characteristic symbol of P . We assume that S is a noncharacteristic surface. Definition 9. The operator P is said to be elliptic at a point 0 E T*X\O if g(0) # 0. The problem (P, B) is elliptic at a point p E T*S\O if P is elliptic on z'p and the map +
c'
(2.77)
is a bijection; in particular it is necessary that dim(Ker p ( p , 0,) n cp(lR+)= p.
Theorem 28. (i) If Pu E d(8), then WF,(u) c C. (ii) If Pu E d ( X )and yBu E d ( S ) , then Cb.
Definition 10. The operator P is said to be microhyperbolic at a point 0* E T*X\O in the direction u = (ux, uc) if there exists a neighbourhood V of this point in IR2"+2 and an E > 0 such that g(O + itv) # 0 V0 E V, t E (0, E). (2.78) E T*S\O. Let zk(k = 1, ...,N ) be the distinct roots of the polynomial C,), where z l , . .., z M E IR and z ~ +. .~., zN , E C\R. Let u = ( u i , 0); E JR2" and Ck E IR ( k = 1, . . .,M ) . Assume that P is microhyperbolic at the points
Let p*
g(p*,
uk
= (p*, z k )in
= (Ui,
0,U;, ( k )
the directions
(k = 1,. .. , M ) .
yb(p
+ ito, D,): E ( p , t ) + C'
(2.80)
is a bijection for sufficiently small I/ and E > 0, then the problem (P, B) is microhyperbolic at the point p* in the multidirection (o, C1, . . .,
cM).
Theorem 29. Let p* E T*S\O. Let zk (k = 1, . . .,M ) be distinct real roots of the polynomial g(p*, z). Suppose that 0: = (p*, zk) E z'p* and V and V, are neighbourhoods of p* and 0* in T*S and T * X , respectively. Let (pk = (pk(x,t') E d(V,)( k = 1, . . ., M ) , cp = cp(x', 5') E d ( V ) be real, positive homogeneous functions of degree 0 such that q1 = . * . = cpM = cp on T*S, cp(p*) = 0 and the problem ( P , B ) is microhyperbolic at p* in the multidirection H,, __ acp1
")
(
( p * ) . If Pu E d ( X ) ,yBu
Ed
ax,
,... ,
( S ) and
ax,
Let T*X\O 2 C and T*S\O 2 Cbbe the set of points where P and (P, B) are, respectively, nonelliptic. We shall call 2 and C b the characteristic set of P and ( P , B), respectively.
0:
Definition 11. If the condition (2.79) holds and if the map

Here, and in what follows, y denotes the restriction operator to {xn = 0).
wFba(u)
+
+
k e Z+
yb(p, Dn):Ker P(P,0,) n~ ( w ' )
219
(2.79)
wF,(u) n V, n { (pk < o} = @ Vk = 1, .. ., M ,
(2.81)
WFba(U)n V n {cp < 0) = Izr,
(2.82)
then p* E wFb,(U). We note the closeness between Theorems 24 and 29. 3.2. The Wave Equation. We examine the wave equation in some more detail. More generally, we shall consider a differential operator P of second order of the form (2.65)with analytic Coefficients. Suppose that either B = I or B is a differential operator of first order satisfying conditions (2.66) and (2.67), with analytic coefficients, or, finally, B = D, + /?(x'), with fi E d ( S ) . Let us consider the problem Pu E d ( X ) ,
yBu E d ( S ) .
(2.83)
Since it is elliptic in 8,its solutions satisfy the inclusion WF,,(u) c C. Definition 12. By an analytic ray we mean a Lipschitz curve y = (0 = 0(t)}, with parameter t = xo, that lies in Z and is either a bicharacteristic or a boundary bicharacteristic in the neighbourhood of all its points.
220
11. Linear Hyperbolic Equations
V.Ya. Ivrii
The exceptional points then constitute a locally finite set on each analytic ray. For 8 E V n 2 and an open set zlzY c V, we denote by K'(8, V ) the union of all analytic rays issuing from in the direction of increasing (decreasing)xo and lying in It can be shown that K:(8, V) possess properties (1.43)(1.46). From Theorem 29, we have
i
Theorem 30. Let u E 9'(@satisfy (2.83), z'zV c V be an open set, and j VnCeC.If ~ WF,,(u)n K i ( 8 , V ) n aV = $3,
then
WFf,(u)nK;(8,
B
We discuss the propagation of singularities by assuming that either E + p. If u is a solution of (2.83), then, by Theorem 30,
ua,)nWF,,(u) = 0=v4wF,a(u). We recall that, by virtue of Theorem 25, (YT
y T nWFJ(u)=
K+
I
S
I+
a+
Fig. 5
Fig. 6
In accordance with the customary terminology, y is an incident ray, y+ a reflected ray, a+ a crawling ray, and for any p E a+ $+ = y+(p) is a glancing ray. Therefore
K : ( n = i j u Y+ u a+ u
u
or
(2.84)
@*p$
WFf(u). (2.85) As shown by a direct computation by Rauch [1977] for the wave equation and by Friedlander and Melrose [1977] for the model equation, the implication (2.85) is not valid for analytic wave fronts. We also have the following results. (7 uy+)nWF,,(u) = $3 * i 5 $ WF,,(U). We note that here we do not require that boundary conditions be satisfied; we also note that the assertion is not valid for CODwavefronts.
Theorem 32 (Sjostrand [1980b, c, 19813). The problem (2.83) has a solution u such that WF,(u) = WF,,(u) = 7,. In particular, (a u a+) n WF,,(u) = 0does not imply that p $ WFf,(u). Theorem 33 (Sjostrand [1980b, c, 19811). if u is a solution of (2.83), then
(Y F u a*) u WFf,(U) = 0* i j $ WFf,(U). In conclusion, we consider precisely the wave equation nu = 0 with n space variables. Let w be an open domain in IR" with analytic boundary ao and convex complement, that is, w is an "exterior" domain. Let y' E w and u a solution of the problem
nn I+
=i
=0 .
Theorem 31 (Kataoka [1981]). I f Pu E d ( X ) ,then
V ) = 0.
What is more, with the aid of Theorems 28 and 29 we can investigate the propagation of analytic singularities of solutions to nonclassical problems. Let us discuss the distinction between the CODraysand the analytic rays. A boundary bicharacteristic is always an analytic ray and it is a C"ray only outside the dijiractive set '3+ = { p E '3, qx,(p) > O}. Therefore through every point of this set there passes (locally) only one CODraybut several analytic rays, and the branching of analytic rays takes place at points of '3+ and at some points of its boundary a??+; at these points transition takes place from tangent bicharacteristic to S to boundary bicharacteristics and viceversa. To see the effects of the fact that an analytic ray through i5 E '3+ is not unique, we let pass through this point a bicharacteristic y and a boundary bicharacteristic a. Put y * = y n { + x o > 0} and a* = a n { f x , > 0 } (we assume that x , @ ) = 0); everything is depicted in Fig. 5 for the wave equation where S is not straight. Then K + ( p ) = i j u y+ but at the same time K:(ij) has a more complicated structure (see Fig. 6). x
22 1
Y+(P).
pea+
Instead of showing K + ( p ) and K:(ij), Fig. 6 shows their intersection with the hyperplane { x o = t } , t > 0.
nu = 0 in IR x w, ul,..=, = 0,
ulRxaw= 0
D , U ~ , , == ~ iS(x'  y').
(2.86) (2.87)
Theorem 34 (Rauch and Sjostrand [1981]). Suppose that either n = 2 and aw is convex or n 2 3 and aw is strictly convex. If u is a solution of (2.86)(2.87), then WF,,(ti) is precisely the union of all the analytic rays issuing from the points (0, y', 1, <'), where 5' runs through IR"n { 15'1 = l}. At the same time, WF,(u) is the union of all Cwrays issuing from these very points. In particular, sing supp, u = .n,WF,,(u) contains the frontal boundary of supp u.
Problems In conclusion, we should like to formulate a number of problems whose solution should prove, in our view, extremely useful. Among these problems there are problems of most diverse degree of difficulties.
V.Ya. Ivrii
11. Linear Hyperbolic Equations
1. Prove the IvriiPetkov conjecture (Q 1.4, Chap. 1) in the general case (that is, when S , u S # 0). 2. Generalize Nishitani's results (Q 1.9, Chap. 1) to the case when S , u S = 0 and to equations of arbitrary order. 3. Is it true that if P is an x,hyperbolic operator, then an operator Q of order not exceeding m  1 can be found such that the Cauchy problem for P + Q is C"well posed? 4. Is it true that if the Cauchy problem for P is C"well posed and I is the maximum multiplicity of characteristics (or of characteristic roots), then the Cauchy problem for P + Q is also C"well posed provided that the order of Q does not exceed m  r? 5. Apart from the strong formulation of the Cauchy problem, which we have used, weak formulation for it is also possible under which we seek the solutions in the spaces 9' and 9i.). To prove the necessity of the wellposedness conditions, we often start from the weak formulation while to prove the sufficiency from the strong formulation, and thereby establishing their equivalence. We should like to have a more general and direct proof of the equivalence of these two formulations. 6. Give a general and direct proof of the fact that under suitable regularity conditions on the coefficients of P the following implications hold
in some spatial type lens, it will remain so in a smaller spatial type lens whose base is a piece of an arbitrary spatially similar surface, and not necessarily the hyperplane {x, = T } ? 11. Prove Theorems 36,37,38,39 and 42, Chap. 1, for all those operators for which it is known that the Cauchy problem is suitably well posed. 12. More generally, give a direct proof of the fact that the wellposedness of the Cauchy problem implies the validity of these theorems. 13. If
222
n
P=
G("I)wellposedness (for x1 c x)
and local G(")wellposedness G(")wellposednessin some neighbourhood of the set X T . This problem will be solved if one solves the preceding problem. 7. Prove Theorem 2, Chap. 1, under the assumption that S , u S = 0. 8. Let p be a critical point of the principal symbol p of a scalar x,hyperbolic operator P . Assume that the fundamental matrix F,(p) does not have nonzero real eigenvalues and that one of the conditions a) or b) of Theorem 18, Chap. 1, is violated. Then it can easily be shown that the Cauchy problem is not G{"}well posed for sufficiently large x. Find the (best possible) value of x. Does x depend on the structure of the fundamental matrix or on the fact which of the conditions a) or b) is violated? What, in particular, is the value of x for the operators Pl + cDn( I = 2, 3) of Example 3? 9. Is the Cauchy problem for the operator P
= D;  x:DI
+ uD,
G(")wellposed? If not, establish a Theorem 33 type (Chap. 1) general result that will imply this result. 10. Is it true that all the wellposedness conditions, except the x,hyperbolicity condition, for the Cauchy problem can be written in a symplectically invariant form? Is it true, in particular, that if the Cauchy problem is well posed
1 uj(x)Dj+ b(x) j=O
C"wellposedness =G(")wellposedness=. G(")wellposedness
223
i
is a symmetric hyperbolic matrix operator, then prove that there exists a nowhere dense closed set A in the space of C"matrices that depends on p and is such that the inclusion (1.51) turns into an equality when b $ A . 14. Similarly, show that if P is x,hyperbolic, we can find an operator Q of order not exceeding m  1 such that the Cauchy problem for P + Q is C"well posed and that for this operator equality (1.51) holds. 15. Obtain necessary and sufficient conditions for the weak &wellposedness of the normal mixed problem for equations of an arbitrary order. 16. Define the notions of a C"ray and an analytic ray for the general mixed problem and reformulate in terms of these Theorems 24,25 and 29, Chap. 2. 17. Investigate the propagation of singularities of solutions of the mixed problem in the Gevrey classes for the wave equation, first of all, and before this for the Freidlander's model problem [1976], and show that for x > x * the G(")singularitiesand for x > x* the G(")singularities are propagated in the same manner as the C"singularities are. Show further that for other values of x, the singularities in Gevrey classes are propagated just as analytic singularities are. And that in this case x* = 3, a fact dictated by the order of decay of solution to the Helmholtz equation ( A + k 2 ) o = S(x'  y') in the shade zone as the frequency k increases.
Bibliographical Remarks
1 1
'
In preparing the bibliography the author was faced with the daunting task of selecting the titles to be included in the list. It suffices to mention that the first version of the list of references was four times as large as the final version, and yet it was far from complete. In selecting the titles, the author was guided by a number of principles. An attempt was made to include the most recent works, or those which remained undeservedly, in the author's view, less known or those whose results were used by the author or, finally, the ones which had a rich bibliography. But at the same time, some works which proved to be useful for the author in writing this paper could not be includedin the list. The author does not consider himself to be competent enough to comment on the historical themes. Therefore these remarks should not be taken either as a historical survey or as an investigation into the question of priority.
224
V.Ya. Ivrii
11. Linear Hyperbolic Equations
We note, first of all, the fundamental works of Hormander [1963, 1983, 19851 devoted to the general theory of partial differential operators, the works of Duistermaat and Hormander [1972], Hormander [1971,1983,1985], Taylor [1981] and Trtves [1980] devoted to the theory of pseudodifferential and Fourier integral operators, and the works of Maslov [1965, 19833 and Maslov and Fedoryuk [1973, 19763 where the theory of canonical operators has been discussed. A large number of works are devoted to the C"well posed Cauchy problem. Here one must mention, first of all, the classical work of Hadamard [1932]. Petrovskij [1938], Leray [1954], and Girding [19571 have established the L,wellposedness of the Cauchy problem for strictly hyperbolic operators. The necessary conditions (but not the sufficient conditions for a matrix operator) for the L,wellposedness have been obtained by Ivrii and Petkov [1974], and Strang [1966,1967]. That hyperbolicity is necessary for the C"wellposedness has been proved by Lax [1957], Mizohata [1961], and Ivrii and Petkov [1974]. Necessary and sufficient conditions for the Cauchy problem to be C"well posed for equations with characteristics of constant multiplicity have been obtained by Flaschka and Strang [1971], Ivrii and Petkov [1974], and Chazarain [1974], respectively. Operators with constant coefficients have been studied by Girding [1951] and Hormander [1963, Chap. 51. These work as well as those by Bronshtejn [1974], Atiyah, Bott and Girding [1970, 19731, John [1978], Nuij [1968], and Svensson [1970] discuss the algebraic properties of hyperbolic polynomials. The necessary conditions for the regular and completely regular hyperbolicity can be found in the work of Ivrii and Petkov [1974], while the sufficient conditions in the works of Ivrii [1976a], Iwasaki [1983, 19841, and Melrose [1983]. The necessary conditions for the C"wellposedness of the Cauchy problem for equations with characteristics of variable multiplicity were obtained by Ivrii and Petkov [1974], Hormander [1977], and Yagdzhyan [1980]; the last work mentioned deals with degenerate equations having gluing of characteristic roots of infinite order. Sufficient conditions under which the Cauchy problem for degenerate equations is well posed have been obtained by Nuij [I9681 and Yagdzhyan [1980], and for fairly general irregular hyperbolic equations by Ivrii [I9771 and Hormander [1977] by the method of energy integral. A complete analysis of equations of second order in two independent variables with analytic coefficients in an open domain has been carried out by Nishitani [1984]; some model equations with characteristics of variable multiplicity 3 or 4 have been investigated by Zajtseva [1980]. Petkov [1975, 19783 and Demay [1977] have obtained necessary and suflicient conditions for the C"wellposedness of the Cauchy problem for systems of first order with characteristics of constant multiplicity. A necessary condition for the regular hyperbolicity of systems in two independent variables has been obtained by Petkov and Kutev [1976]. A rich bibliography on the C"well posed Cauchy problem can be found in the works of Ivrii and Petkov [1974], Girding [1982], and Iwasaki [1983]. Not too many works have been devoted to the wellposedness of the Cauchy problem in Gevrey classes. The necessity of the hyperbolicity has been established by Ivrii [1976b], Komatsu [1977], and Nishitani [1978], while sufficiency for the wellposedness in arbitrary Gevrey classes by Ivrii [1975], Bronshtejn [1980], and Kajitani [1983]. The necessary and sufficient conditions for wellposedness in Gevrey class with suitable exponents for equations with characteristics of constant multiplicity have been obtained by Ivrii [1976b], and Komatsu [1977], while the paper by Ivrii [1976c] contains the necessary conditions for the wellposedness of equations with characteristics of variable multiplicity. Some degenerate equations have been dealt with by Ivrii [1978], and Yagdzhyan [1978]. Ivrii [1975, 1976b, 19783 and Nishitani [1978] have obtained the energy estimates in Gevrey classes. For a discussion of operators with constant coefficients, we once again refer the reader to Hormander [1963, Chap. 51. A large number of works have been devoted to the study of the propagation of singularities of solutions to the Cauchy problem and the related propagation in the whole space. One of the earliest works in this direction is by Babich [1960], where a fundamental solution for the strictly hyperbolic equation has been constructed, and by Lax [1975], where the propagation of oscillations has been described; a still earlier work of Courant and Lax has been overshadowed by Babich [1960]. The remaining works in our list are recent. Atiyah, Bott and Girding [1970, 19733 discuss the sin@larities of the fundamental solutions of equations with constant coefficients. Duistermaat and Hormander [19723 describe the propagation of singularities of principal type equations by meam of the Fourier integral operators. Parametrices for the Cauchy problem for operators with c h a m 
teristics of constant multiplicity have been constructed by Chazarain [I9741 and by Petkov [1978], respectively, for equations and for firstorder systems. Alinhac [1978], Yoshikawa [1977, 1978a,b, 19801, and Kucherenko and Osipov [1983] have constructed parametrices for various classes of effectively hyperbolic operators. Nishitani [1983] describes the propagation of singularities for general effectively hyperbolic operators; close results have been obtained by Ivrii [1979a]. The propagation of the singularities of the solutions to various classes of equations has been discussed by Ivrii [1979a], R. Lascar [1981], Melrose and Uhlmann [1979a,b], and Uhlmann [1982]. For a description of the propagation of singularities of solutions to symmetric hyperbolic systems of first order, the reader is referred to Ivrii [1979c]; all these results have been reformulated by Wakabayashi [19801 in the language of generalized bicharacteristics. A detailed discussion of hyperbolic pseudodifferential systems that are principally diagonal can be found in Kucherenko [1974], R. Lascar [1981], Lax [1957] and Ralston [1976]. All these works concern the propagation of C"singularities. The works dealing with the propagation of singularities in Gevrey classes are few, the work by Wakabayashi [1983] being the most important. From among a significant number of works devoted to the propagation of analytic singularities, we mention only Wakabayashi [1983], where the results of Sato, Kashiwara and Kawai have been reformulated in terms of the generalized bicharacteristics. Of works concerning the wellposedness of the mixed problem, we must mention, first of all, the works of Schauder, of Krzyzanski and of Ladyzhenskaya that are devoted to strictly hyperbolic equations of second order, and the works of Friedrichs [1958] and of his students Lax and Phillips [1960] that are devoted to positive (that is, strictly dissipative) problems for symmetric systems. Of the more recent works, we note Sakamoto [1970a,b, 19801, and Kreiss [1970], Agranovich [1969, 19721, Ralston [1971] and Rauch [1972], where strong L,wellposedness of the mixed problem has been investigated, respectively, for equations and firstorder systems. The weak L,wellposedness has been dealt with by Agemi and Shirota [1970, 19713, and by Miyatake [1975,1977]. Miyatake's papers contain almost exhaustive results for secondorder equations. A necessary condition for the C"wellposedness under very general condition can be found in Wakabayashi [1980]. The C"wellposedness of the mixed problem for the wave equation has been analysed by Hormander [1983, 19851 and Soga [1979, 19803. The general problems for strictly hyperbolic equations of second order have been studied by Eskin [1979/80, 1981a,b]. A nonstrictly model hyperbolic problem has been discussed by Zajtseva [1983]. Under fairly weak assumptions, it has been shown by Duff [1959] that the mixed problem is analytically well posed. Girding's [I9821 survey paper should be mentioned once again. Povzner and Sukharevskij inagurated the study of the propagation of C"singularities of the solutions to the mixed problep. Chazarain [1977] and Nirenberg dealt with the same question in the language of wave fronts, but they examined only transversal reflection of waves from the boundary. For this case, a parametrix has been constructed by Fedoryuk [1977]. The propagation of singularities in bicharacteristically strictly concave domains has been investigated by Friedlander [1976], Melrose [1975, 19781, Taylor [1976a,b, 19791, Eskin [1976], while in bicharacteristically convex domains by Ivrii [1979d,e, 19801, Eskin [1977]; the domains of an arbitrary nature have been dealt with by Ivrii [1980b], Melrose and Sjostrand [1978, 19823; for investigations concerning very general symmetric systems, see Ivrii [1979d,e 1980a,b, 19863. The propagation of singularities of solutions to the nonclassical problems for strictly hyperbolic equations of second order has been examined by Ivrii [1981], Eskin [l979/80,1981a,b], Melrose and Sjostrand [1978,1982]. Very few works deal with the question of the propagation of analytic singularities. In this direction, one must mention, first of all, a very general work of Sjostrand [1980]. Friedlander and Melrose [1977], Kataoka [1981], Rauch and Sjostrand [1981] and Sjostrand [1980b,c, 19813 have investigated the propagation of analytic singularities for strictly hyperbolic equations of second order. The maximum attention in these works is directed to bicharacteristically concave domains where the propagation of C"singularities and that of analytic singularities are significantly different. In this very situation, Rauch [I9771 examined the propagation of the frontal boundary of the support of the solution, the propagation being closely related to that of analytic singularities.
,
i
i lI
I
1 i I
225
11. Linear Hyperbolic Equations
V.Ya. Ivrii
226
A Survey of Recent Results
227
Of the works on the propagation of singularities of solutions to the Cauchy problem, we mention the one by Melrose [1986]. This paper examines the propagation of singularities for the equation u,, Au = 0, where A is a selfadjoint positive definite differential operator of the second order and is hypoelliptic with a loss of 2  6 derivatives, with 6 > 0. A number of works deal with the propagation of the polarization front. Namely, if u E g ( X , CN), then (x, <,w ) E T * X x C N lies in WF;,,(u) provided that for any pseudodifferential operator A of order 0 the inclusion Au E H" implies that a(x, < ) w = 0, where a is the principal symbol of A. The wave front is a projection onto T*X of the polarization front. The concept of the polarization front was introduced by Dencker [1982], and his investigation concerns the propagation of polarization fronts of solutions to systems of real principal type with characteristics of constant multiplicity. It turns out that for such systems the polarization fronts are propagated along Hamiltonian orbits, namely, curves lying in T*X x (CN and determined by a system of ordinary differential equations; their projections onto T*X are bicharacteristics. Dencker [1986] studies the propagation of the polarization fronts for systems with conical refraction, and in this study the distinction between the real polarization and the complex polarization plays a significant role. Some results also exist concerning the propagation of the polarization fronts near the boundary, namely, concerning their reflection under transversality of bicharacteristics and even diffraction. Some results regarding the analytic polarization fronts have also been obtained (see the bibliography in Esser [1986]). Many works study mixed problems. Eskin [1985] has investigated the wellposedness of the mixed problem for a strictly hyperbolic operator of second order in a bicharacteristically convex domain and has generalized his results of [1981b]. For this problem, a parametrix has been constructed by Kubota [1986] under certain conditions imposed on the boundary operator and the standard results on the propagation of singularities have been obtained. Laubin [1986] has given new proofs for Theorems 32 to 34 of Chapter 2 of the present survey. Zajtseva [1987a] has obtained a condition on (a, a) E (C u co)x C that is necessary and sufficient for the mixed problem
+
The main part of the paper was written in 1984; however, the works published during the past three years' have made this supplement necessary. We mention, first of all, the surveys by Girding [1984] and by Mizohata [1986a]. The book Mizohata [1986b], where the latter survey has been published, contains a number of important works some of which have been included in our reference list. We note that at last we have in the Russian language the main works of Petrovskij; in particular, the works on the hyperbolic equations and systems (see Petrovksij [1986]). A number of works have been devoted to the C"wellposedness of the Cauchy problem. Nishitani [1985, 19863 has obtained the energy estimates for effectively hyperbolic as well as for some other operators with double characteristics, and has given an alternative proof for the C"wellposedness of the Cauchy problem for effectively hyperbolic operators. Yagdzhyan [1985,1986] has found necessary and sufficient conditions for the Cauchy problem to be C"well posed for degenerate operators having gluing of characteristic roots of infinite order, and has also constructed a parametrix for the Cauchy problem. Zajtseva [1987b] has shown that in order that the Cauchy problem for the operator
p = 0:  0:  x2"D; + x"' DY be C"well posed for odd n, it is necessary and sufficient that a E R, la1 < n. It has been recently discovered that the condition a E R u iR is necessary and sufficient for the problem to be C"well posed for even n provided that la1 is sufficiently small. Nishitani [1985] has studied the firstorder systems and has shown that if P is a matrix operator of first order with no more than double characteristics, then for the strong (regular) hyperbolicity of P, it is necessary that in a neighbourhood of every point p either the principal symbol p is uniformly diagonalizable or g = det p is effectively hyperbolic. If both these conditions are violated at p E T*R"+'\O,then for the Cauchy problem to be C"well posed, it is necessary that there exists an a E [  1, 13 and a matrix B such that i (pscop  ? { p , '"p)
+ a Tr'g
I
+
'The original Russian edition of the book was submitted in 1987 and published in 1988.Translator
0 :
 ~ ' 0 ,+' aDy, 0,
+ aD,)
to be well posed. Finally, Lebeau [1984, 19861 has shown that for the Dirichlet problem for a strictly hyperbolic operator of second order with analytic coefficients in a domain having analytic boundary, the G%ingularities, for x 2 3, are propagated in the same manner as the C"4ngularities are, while for 1 < x < 3 in the same way as the analytic singularities are (recall that analytic singularities can be propagated along crawling rays and glancing rays but C"singularities cannot). Thus, Problem 17, formulated above, has been partially solved.

Conversely, if in the neighbourhood of every point either g is effectively hyperbolic or p is uniformly diagonalizable and if ,Z = {g = dg = 0) is a C"manifold of codimension d < 3, TpZ = Ker F,(p), then P is a regularly hyperbolic operator. A number of works deal with the wellposedness of the Cauchy problem in Gevrey classes. Kajitani [1986] has established that under suitable regularity conditions on the coetlicients, the Cauchy problem for a uniformly diagonalizable system of first order is G(")wellposed for x < 2. The results of Bronshtejn [1980] have been generalized by Ohya and Tarama [1986] to the case where the coefficients are not very smooth with respect to xo. Namely, they have shown that if the with u E (0,2], coefficients of the principal part of the scalar operator P lie in C'([O, TI, G(")(R")), and the lower terms lie in Co([O, TI, G(")(R")), then the Cauchy problem will be G(")well posed if 1 < x < min(1 u/r, r/(r  l)),where r is the maximum multiplicity of the characteristic roots. The Italian mathematicians have obtained considerably more precise results for secondorder operators, and very often they have examined operators without lower terms and with coefficients depending only on x,,. They have succeeded in discovering a large number of illuminating counter examples. In particular, it has been shown that for a strictly hyperbolic operator, with u < 1, x > 1/(1  a), the Cauchy problem, in general, is neither G(")wellposed nor C"well posed. Moreover, for u < 1, even the uniqueness theorem may fail to hold. A survey and bibliography can be found in Spagnolo [1986] and Colombini, Jannelli and Spagnolo [1986].
{D:
I
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Construction of a parametrix for the Cauchy problem of some weakly hyperbolic equation. 111. ibid, 127141,Zb1.393.35040 [1980] Parametrices for a class of effectively hyperbolic operators. Commun. Partial Differ. Equations 5, No. 11, 10731151,Zb1.483.35048 Zajtseva, O.V. [1980] On the wellposedness of the Cauchy problem for a model equation with quadruple characteristics. Izv. Vyssh. Uchebn. Zaved., Mat. 1980, No. 6, 2022. English transl.: Sov. Math. 24, No. 6, 2123 (1980), Zb1.441.35041 [1983] On the wellposedness of a model nonstrictly hyperbolic mixed problem. ibid. 1983, No. 10, 2225. English transl.: ibid. 27, No. 10, 2832 (1983),Zb1.599.35094 [1987a] On the wellposedness of a new model nonstrictly hyperbolic mixed problem. ibid. 1987, No. 11, 1012. English transl.: ibid. 31, No. 11, 1215 (1987),Zb1.679.35060 [1987b] The wellposedness conditions for the Cauchy problem for a certain model nonstrictly hyperbolic equation. ibid. 1987, No. 12, 4345. English transl.: ibid. 31, No. 12, 5355 (1987),Zb1.672.35038
Author Index Agemi, R. 201,225,227 Agranovich, M.S. 15, 154,225,227 Airy, G.B. 24,203,215 Alinhac, S. 189,225,228 Arnol’d, V.I. 10, 20,46, 139 Atiyah, M.F. 32, 36, 158, 160,224,228 Babich, V.M. 224,228 Baire, R. 111, 185 Banach, S. 77 Beals, R. 21,26, 29, 88,95, !00,107, 139 Bessel, F.W. 77 Bohr,N. 49 Bokobza, J. 15,26,27, 147 Bolley, P. 100, 139 Bony, J.M. 83, 133, 139 Bott, R. 36, 158160,224,227 Boulkhemair, A. 83, 139 Boutet de Monvel, L. 98100, 116,118, 139 Bove, A. 81, 140 Bronshtejn, M.D. 100, 140,161,175,224,226, 228 Bros, J. 189 Calderon, A.P. 14,25, 140, 157 Campbell, J.E. 106 Camus, J. 100, 139 Cauchy,A.L. 22,34,43,44,50, 115, 119, 132, 151158,168,169,173,177,222224,226, 227 Chazarain, J. 83, 140, 167, 189,224,225,228 ChinHungChin 25 Colombini, F. 226,228 Courant, R. 224 Demay, Y. 173,224,228 Dencker, N. 140,227,228 Derridj, M. 114, 140 Dirac, P.A.M. 54 Dirichlet, L.P.G. 116 Doughs, A. 108, 112 Duff, G.F.D. 206,225,228 Duhamel, J.M.C. 198 Duistermaat, J.J. 76, 78, 80, 140, 180, 224, 228 Dynin, A.S. 15
Egorov, Yu. V. 7, 10, 11, 20,23, 75, 8789, 100, 102,104,140 Eskin, G.I. 15, 141, 204206, 215, 225, 227, 228 Esser, P. 227, 229 Farris, M. 141 Fedoryuk, M.V. 42,53,144,224,225,229, 231 Fedosov, B.V. 32,141 Fefferman, C. 26,88,100,107,139,141 Flaschka, H. 82,141,166, 167,224,229 Folland, G.B. 100,141 Fourier, J.B.J. 8, 9, 32,4346, 50, 52, 55, 56, 58,59,6165,71,72,102,127,128, 138 Friedlander, F.G. 141,221,223,225,229 Friedrichs, K.O. 158, 198, 229 Frobenius, F.G. 79,105 Ganzha, E.I. 100, 107,141 Girding, L. 18, 120,141,157160,224226, 228,229 Gerard, C. 141 Gevrey, M. 10,91,92, 114, 173, 174, 177, 189, 223,224 Giraud,G. 14 Green,G. 75 Grigis, A. 83, 99, 100, 140142 Grubb,G. 142 Grushin, V.V. 26,84,96100, 113,142,147 Guillemin, V. 11, 140, 142 Hadamard, J. 115,224,229 Hahn,H. 77 Hamilton, W.R. 40 Hanges, N. 81, 142 Harvey, R. 74,75, 142 Hausdorff, F. 74, 106,111 Heisenberg, W. 8,40 Helffer, B. 84,99, 100,139,140, 142 Helmholtz, H.L.F. 42,223 Hilbert, D. 33 Holmgren, A. 134,196 Hormander, L. 7, 10, 12, 13, 15, 17, 18, 24, 25, 27, 29, 31, 32, 51, 53, 54, 59, 61, 64,65, 73, 76,
238
Hormander, L. (cont.) 78,80,85,9195,99, 100, 102, 106,109, 140, 142, 158160, 163, 168, 175, 180, 188, 189, 224,225,228,229
Lipschitz, R. 65 Lopatinskij, Ya. B. 195, 197, 199 Ludwig, D. 215 Lychagin, V.V. 144
Iagolnitzer, D. 189 Ikawa, M. 203,204,229 Iordanov, LV. 145 Ivrii, V. Ya. 7, 39, 81, 83, 143, 155, 156, 158, 162165, 167169, 174, 175,177179,181, 184, 186, 188, 192, 199, 209, 210, 212, 213, 217,224,225,229,230 Iwasaki, N. 164,168,224,230
Magenes, E. 75, 144 Mandai 169 Martineau, A. 129, 132, 144 Maslov, V.P. 20,42,46,50,53, 144, 154,188, 224,23 1 Melin, A. 71, 144 Melrose,R. 39,81,120,121,126,144,164,189, 203,208,214,215,217,221,224,225,227,229, 232 Menikoff, A. 83,99, 100, 144 Metivier, G. 100, 145 Mikhlin, S.G. 15 Mischenko, A.S. 73, 145 Miyatake, S. 202,203,205,232 Mizohata, S. 84, 153, 156,157,224,226,232
Jacobi, K.G.J. 12 Jannelli, E. 226,228 John, F. 160,224,230 Kajitani, K. 224, 226, 230 Kannai, Ya. 85,99,100,107,143 Kashiwara, M. 79,134, 143, 146,225 Kataoka, K. 221,225,231 Kawai, T. 79, 143,146,225 Kirchoff, G.R. 152 Kohn, J.J. 15,25, 143 Kolmogorov, A.N. 105 Komatsu, H. 174, 177,224,231 Korn, A. 14 Kovalevskaya, S.V. 206 Kreiss, H.O. 199, 225, 231 Krzyzanski, M. 225 Kubota, K. 227,231 Kucherenko, V.V. 73, 143,189,225,231 Kumanogo, H. 26,27, 143, 189,231 Kutev, N.D. 173,224,233 Ladyzhenskaya, O.A. 225 Landis, E.M. 115, 143 Laplace, P.S. 90, 115, 194 Lascar, B. 83, 143,144 Lascar, R. 83,143, 144, 189,225,231 Laubin, P. 227,231 Laurent, Y. 144 Lax, P. 153, 156,198,224,225,231 Lebeau, G. 227,231 Lebesgue, H.L. 40 Leibniz, G.W. 16 Leray, J. 153, 157,224,231 Levi. E.E. 81, 166, 176 Lewis, J.E. 81, 140 Lewy,H. 84 Lichtenstein, L. 14 Lie, M.S. 105, 107, 112, 114 Lions, J.L. 75, 144
Author Index
Author Index
Nazajkinskij, V.E. 73, 145 Neumann, K.G. 98, 102,201 Newton, I. 40,171 Nirenberg, L. 15,25,87, 88, 108, 112, 145,225 Nishitani, T. 170, 174,224226,232 Nourrigat, J. 100, 142 Nuij, W. 160,224,232 Ohya, Yu. 226,232 Olejnik, O.A. 107, 108, 111, 112, 115, 143, 145,170,233 Oshmyan, V.G. 145 Osipov, Yu. V. 189,225, 231 Parenti, C. 81, 100, 140, 145 Parseval, M. 45 Petkov,V.M. 155,156,158,162165,168, 169, 172, 173, 189, 192, 199,224,225, 230, 233 Petrovskij, I.G. 153, 157,224, 226,233 Phillips, R.S. 198, 225, 231 Phong, D.H. 141 Planck, M. 40 Poisson, S.D. 20,40,41, 79, 107, 116, 117, 163 Polking, J. 74, 75, 141 Popivanov, P.R. 87,90,100,113,141,145 Povzner, A. Ya. 225 Puiseux, V.A. 171 Radkevich, E.V. 107, 108,111, 112, 145 Radon,J. 9 Ralston, J.V. 225,233 Rangelov, Ts. 87, 141
Rauch, J. 221,225,233 Rempel, S. 118, 145 Riemann, W. 36,74, 115 Rodino, L. 100, 145 Rothschild, L.P. 100,145 Sakamoto, R. 194,196, 197,225,233 Sato, M. 23,79,129, 146,189,225 Schapira, P. 83, 129, 135, 143, 146 Schauder, J.P. 225 Schrodinger, E. 42,49,50,83, 154 Schulze, B.W. 118, 145 Schwartz, L. 27, 146 Seeley, R.T. 15, 32, 35,37, 38, 146 Segala, F. 145 Seidenberg, A. 91 Shatalov, V.E. 145 Shirota, T. 201,225,227 Singer, I.M. 21, 32 Sjostrand, J. 23, 71, 73, 8183, 131, 141, 144, 146,189,214,217,225,232,233 Sobolev, S.L. 26, 64,73 Soga, H. 203,204,225,233 Spagnolo, S. 226,228,234 Shananin, N.A. 146 Shubin, M.A. 7,32,37, 146 Stein, E.M. 100, 145 Sternberg, S. 11,142 Sternin, B. Yu. 23, 145, 146 Strang, G. 82, 141, 158, 166, 167,224,229, 234 Sukharevskij, LV. 225 Svensson, L. 159,224,234
239
Taniguchi, K. 26,27,189,231 Tarama, S. 226,232 Tarski, A. 91 Tartakoff, D.S. 146 Taylor, M. 7, 73, 146, 166, 180, 181, 215, 224, 225,234 Trtves, F. 7, 10,62, 72, 73,87, 88,98100, 109, 110, 140,145,146,166,180,224,234 Tulovskij, V.N. 82, 146 Uhlmann, G. 83,142,146,188,189,225,232, 234 Unterberger, A. 15,26,27, 147 Vaillancourt, R. 25, 14 Vasil'ev, D.G. 39,147 Vishik, M.I. 15, 26, 147 Volevich, L.R. 153 Wakabayashi, S. 183, 190,200,225,234 Wasow, W. 99,147 Weierstrass, K. 167, 170, 171 Wenston, P. 147 Weyl, H. 18,29, 41 Yagdzhyan, K.A. 170,180,224,226,234 Yamaguti, K. 157 Yoshikawa, A. 189,225,234 Zajtseva, O.V. 224227,234 Zuilly, C. 114, 140 Zygmund,A. 14
Subject Index
Subject Index
, effectively hyperbolic 164 , elliptic 17
quantization 40
, elliptic at a point 218
ray, analytic 219 ,C" 214 , crawling 220 , generalized 216 , glancing 220 , incident 220 , reflected 220 removable singularity 74
, formally adjoint 17 , Fourier integral 43, 150 , Fourier integral operator with complex phase function 65
, Girding hyperbolic in direction N
158 158 , hyperbolic at a point x in direction N 156 , hypoelliptic 13 , integrodifferential 15 , Maslov canonical 46 , microhyperbolic 210 , microlocally hypoelliptic 13  of constant strength 94 , partially hypoelliptic 92 , Poisson 117 , principal type 22 , pseudodifferential 15 , real principal type 87 , regularly hyperbolic 158 , simplest integral 14 , smoothing 16 , strictly x,hyperbolic 156 , subelliptic 101 , variable order 26  with characteristic of constant multiplicity 166 with characteristic roots of constant multiplicity 166 , Weyl 18,29 , x,hyperbolic 156
, Girding x,hyperbolic
atlas, canonical 47 bicharacteristic billiards 21 1 212  C"boundary 212 concave 212  generalized 182 limiting 185  polygonal 186 regular 185 sheet 186 boundary, bicharacteristically nondegenerate 215 , uniformly characteristic 194 boundaryvalue problem, normal 193  boundary
Cauchy problem, Gi.1 regular 178 178 compatibility condition of order s 197  condition of infinite order 191 composition of Fourier integral operators 59  of pseudodifferential operators 16 condition, IvriiPetkovHormander 168 , Levi 81 , generalized Levi 176 , LeviStrangFlaschka 166 , Lopatinskij 200 , uniform Lopatinskij 197, 199 cone, dual 160, 182 coordinates, local canonical 47 ,local
,flat 65 , nondegenerate phase 51  of canonical transformation, generating 20 , operator phase 63 phase 51, 167, 171 , regular positive type 68 ,weight 26 Girding inequality 18 glancing surface 24 Gevrey class 91 , inductive 174 , projective 174 Hamiltonian field 163 hyperfunction 130 index, Maslov 47
, wellposedness 162 integral, oscillatory 50 lacuna of distribution 119 Lagrange manifold 46 subspace 46 lens, spatial type 156 localization at a point 160 Lopatinskij determinant 195 manifold, almost analytic 66
, bicharacteristic 188 matrix, fundamental 163
dissipative boundary operator 209 distribution, Fourier 55 , hypoelliptic 93 , invertible 93 domain, bicharacteristically convex (concave) 205
, Levi 102 , smoothly diagonalizable 158
energy estimate 157 equation, characteristic 156 , degenerate hyperbolic 169 , eikonal 167, 171 , Schrodinger 42
Neumann %problem 102 neighbourhood, conical 17
function, almost analytic 66
, symmetrizable 158 , uniformly diagonalizable 158 microlocality 17 microfunction 135
order of operator 15 operator, analytically hypoelliptic 108 , classical pseudodifferential 15 , completely characteristic 124
24 1
parametrix 18 phase, complex 65 point, critical 163 , diffraction 126 , elliptic 204 , gliding 126 , hyperbolic 204 ,tangent 204 Poisson brackets 20,40, 163 kernel 118 proper map 11 pseudolocality 17 pullback of distribution 12 pushforward of distribution 12
set, characteristic
181, 209
, diffractive 220 , elliptic 204 , stationary 187
, tangent 204 stationary phase method 45 symbol 15 boundary 212 , characteristic 155 , principal 15,155 , real principal type 185 , subprincipal 19, 164 symplectic form 163 transformation, boundary canonical
125
, canonical 20 ,contact 138 , FourierBrosIagonitr transmission 115
190
uniformly characteristic surface 194 wave front 9,180 ,boundary 208 ,boundary of order s 208 ,complete 208  oforders 180 wellposedness of Cauchy problem 153 __ ,C" 155  G("1 153, 174 ,local G I x ) 174 _  _  , L , 156,157 wellposedness of mixed problem 190 _ _ _  ,C" 191 _  _  ,L , (strong) 196 _ _ _  ,L , (weak) 201