Regularisation of Mixed Boundary Problems - publish.UP

Regularisation of Mixed Boundary Problems B.-W. Schulze

A. Shlapunov January 24, 2000

N. Tarkhanov

Supported by the Max-Planck Gesellschaft and the RFFI grant 99{01{00790.

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B.-W. Schulze, A. Shlapunov, and N. Tarkhanov

Abstract

We show an application of the spectral theorem in constructing approximate solutions of mixed boundary value problems for elliptic equations.

Regularisation of Mixed Boundary Problems

Contents

Introduction 1 Selfadjoint operators 2 Bounded operators 3 Green function 4 A crack problem 5 Adjoint operator 6 The case of constant coe cients 7 Iterations 8 Cauchy problem 9 Zaremba problem 10 Examples References

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4 8 11 12 15 19 24 26 29 30 33 35

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Introduction When studying boundary value problems for solutions of an elliptic dierential equation Pu = f , P being of type E ! F for some vector bundles E and F , one uses any left parametrix of P to reduce the problem to the boundary. A powerful tool for such a reduction is the Green formula for P which brings together the values of Pu in a domain D and the Cauchy data t(u) of u on the boundary of D to present u in D modulo smoothing operators. In case P has a left fundamental solution , the Green formula reads u = Pdlt(u) + PvPu in D, where Pdlu0 and Pvf are the double layer and volume potentials corresponding to . If moreover is a right fundamental solution, then the potential Pdlu0 satis es the homogeneous equation Pu = 0 in D, and so the Green formula provides us with a soft version of the Hodge decomposition. However, right fundamental solutions are available only for those P which are determined, i.e., bear as many scalar equations as the number of unknown functions. For overdetermined elliptic operators P , the construction of kernels with the property that P Pdlu0 = 0 in D, for each density u0, is a signi cant problem. In complex analysis such Green formulas are known as Cauchy-Fantappie formulas. They give rise to explicit solutions of the inhomo = f on strongly pseudoconvex manifolds geneous Cauchy-Riemann system @u N or domains in C . Consider the iterations PdlN t(u), for N = 1 2 : : :, in any function space H invariant under Pdl. If is a right fundamental solution of P , then these iterations stabilise because P Pdlt(u) = 0 in D and so PdlN t(u) = Pdlt(u) for all N . In general, since Pdl is the identity operator on solutions of Pu = 0, one may conjecture that the iterations converge to a projection of H onto the subspace V1 of H consisting of u satisfying Pu = 0 in D. Were such the case, the equality 1 X u = V1 u + Pdl Pv Pu (0.1) =0

for u 2 H , would give us a substitute for the Hodge decomposition. This idea goes back to the work of Romanov Rom78] who studied the iterations of the Martinelli-Bochner integral in C N , N > 1. His results were extended in a beautiful way to arbitrary overdetermined elliptic systems by Nacinovich and Shlapunov NS96]. To handle the iterations of Pdl, the idea is to present this integral as a selfadjoint operator on a Hilbert space H . While H is speci ed within the Sobolev space H p in D, p being the order of P , the Hermitian structure of H is dierent from that induced by H p. If moreover 0 Pdl I , then the iterations of Pdl converge, by the spectral theorem, to the projection of H onto the eigenspace of Pdl corresponding to the eigenvalue 1. This eigenspace in


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turn coincides with the null-space of PvP in H , and hence with the null-space of P if Pv is identi ed with the adjoint of P in H . The choice of H is actually suggested by the restriction of to the closed domain D . If Pvf meets automatically some conditions on the boundary of D, for each f 2 L2(D F ), these should be incorporated in the de nition of H . To introduce the relevant Hermitian structure in H , we choose any domain 0 D with C 1pboundary, such that D D0, and we x an extension operator 0 e : H ! H (D E ) with the properties that t (e(u)) = 0 on @ D0, for each u 2 H , and e(Pvf ) = Pvf , for each f 2 L2(D F ). In this way we actually reduce the problem to that on a compact surrounding manifold. As but one example of e(u) we show the solution of the Dirichlet problem for the Laplacian = P P in D0 n D with data t(u) on @ D and 0 on @ D0. De ne a scalar product in H by h(u v) = (P e(u) P e(v))L2(D F ) (0.2) then the identity h(Pvf v) = (f Pv)L2(DF ), for every v 2 H , is immediate. It in turn implies the selfadjointness of Pdl with respect to (0.2). In NS96], H is the whole space H p(D E ), and Pvf = G P (D f ) where G is the Green function of the Dirichlet problem for in D0 and D is the characteristic function of D. In this setting, the orthogonal decomposition (0.1) applies to constructing approximate solutions for the inhomogeneous equation Pu = f and the Dirichlet system for the Laplacian in D. This work was intended as an attempt to develop the approach of NS96] to derive approximate solutions of the generalised Zaremba problem in D. It consists of nding a function u 2 H p(D E ) satisfying u = f in D, whose `tangential part' t(u) takes prescribed values on a part of @ D while the `normal part' n(Pu) takes prescribed values on the complementary part @ Dn . Since the Dirichlet problem in D is elliptic, we may assume without loss of generality that t(u) = 0 on . This causes H to be a subspace of H p(D E ) whose elements satisfy t(u) = 0 on . Moreover, G should be the Green function of the Dirichlet problem in the domain with crack D0 n . Thus, the study of the Zaremba problem leads to the Dirichlet problem for the generalised Laplace operator in a domain with cracks. Note that both mixed boundary value problems and crack problems can be treated in the framework of any calculus on manifolds with edges of codimension 1 on the boundary. At present, several calculi on such singular con gurations are known, namely those by Vishik and Eskin (cf. Section 24 in Esk73]), Maz'ya and Plamenevskii (cf. Chapter 8 in NP91]), Schulze Sch92], Duduchava and Wendland DW95], Rabinovich, Schulze and Tarkhanov RST98], etc. Either of these calculi is applicable. The important point to note here is the function spaces used as domains for pseudodierential operators involved. These are scales of weighted Sobolev spaces H s (D E ) parametrised by s 2 R, where s speci es the smoothness 0

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and stands for the weight. While the spaces may dier from each other in various calculi, they usually coincide with L2(D E ), for s = 0. As weight functions, one uses the powers of the regularised distance to the set of singularities, here to the edge @. Moreover, we may always specify the weight exponent in such a way that admissible operators of order m map H s (D E ) to H s;m;m (D F ). Let us dwell upon the contents of the paper. Section 1 presents generalities on bounded selfadjoint operators in Hilbert spaces. In Section 2 we indicate how these results apply to non-necessarily selfadjoint operators. The idea is to replace the equation Bu = f by B Bu = B f and shrink the domain of f to those which are orthogonal to the null-space of B . Section 3 discusses in detail the construction of the Green function for the classical Dirichlet problem in a plane domain with a crack along a segment. We make use of layer potentials to present the Green function, thus showing its asymptotics near the boundary points of the segment. This section is intended to motivate our investigation of crack problems. In Section 4 we formulate the Dirichlet problem for the generalised Laplacian in a domain with a crack along a hypersurface with C 1 boundary. When compared to general crack problems, the Dirichlet problem has the peculiarity of being formally selfadjoint. Moreover, any solution of the corresponding homogeneous problem is trivial unless it is rather singular. Hence it follows that the Dirichlet problem is uniquely solvable provided it is Fredholm. Since the Dirichlet problem is elliptic in the usual sense on the smooth part of the boundary, i.e., away from @, the Fredholm property of this problem is equivalent to the bijectivity of an operator-valued symbol called the edge symbol. This symbol lives in the cotangent bundle to @ with the zero section removed. In fact, the edge symbol is the Dirichlet problem for a dierential operator of special form in the plane with a cut along the non-negative semiaxis. The operator is obtained from P P by freezing the coecients at any point y 2 @ and substituting the covariable for the derivatives along the edge. It is supplied by the Dirichlet boundary conditions from both sides of the cut. Using polar coordinates in the normal plane to the edge at y, we see that the conormal symbol of the edge symbol is (p(P )) p(P ), the principal symbol of P being evaluated at = 0 and

1 = cos 'Dr ; (1=r) sin 'D' 2 = sin 'Dr + (1=r) cos 'D' 1, 2 standing for the conormal variables. Thus, if supplied by the Dirichlet conditions at the endpoints of 0 2], the conormal symbol belongs to the same class of boundary value problems, now on a segment. The condition of bijectivity of the conormal symbol gives us a discrete set of \prohibited"


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weights , independent of . For 62 , the edge symbol is a Fredholm operator in Sobolev spaces of weight over R2 n R +. It is to be expected that the null-space of the edge symbol is trivial if 0 is large enough. By duality, if ; 0 is large enough, then the cokernel of the edge symbol is trivial. It is now the property of P , t and the singular con guration whether or not, given a , the edge symbol is an isomorphism for all 6= 0. For the Dirichlet problem in a domain with a smooth edge, the edge symbol is known to be an isomorphism for in an interval around p (see Sections 6.1.3 and 8.4.2 in NP91]). It follows that the Dirichlet problem for the Laplacian = P P is solvable in H sp(D E ), D being a domain with smooth edges. Then, we make use of a familiar classical scheme to construct the Green function of the problem. Section 5 contains a construction of a scalar product h(u v) in the space p pp H (D E ), such that the norm h(u u) is equivalent to the original one. If restricted to the subspace H of H pp(D E ) which consists of the functions with vanishing Dirichlet data on , this scalar product makes selfadjoint the double layer potential related to the Green function of a larger domain D0 with a crack along . In Section 6 we simplify the construction of h(u v) in case P is a dierential operator with constant coecients in Rn. If P P has a fundamental solution of convolution type G (x) which decreases at in nity fast enough, we may think of G (x) as a Green function of the one-point compacti cation of Rn. Moreover, we can solve the Dirichlet problem for G (x ; y), y 62 , with data on , thus arriving at a Green function of Rn n . Hence we have a canonical choice for D0, namely D0 = Rn. This works, in particular, for rst order homogeneous operators P with constant coecients. Section 7 provides a detailed exposition of iteration of the double layer and volume potentials. Let A be a non-negative selfadjoint bounded operator in a Hilbert Rspace H , satisfying kAk 1. By the spectral theorem, we N have AN = 01+0 ; dI () for any N = 1 2 : : :, where I () is a resolution of the identity in H corresponding to A. Hence it follows that the iterations of A converge to the orthogonal projection of H onto the null-space of the complementary operator I ; A. As but one application of this we show the orthogonal decomposition 1 X N+ A (I ; A) I = Nlim A !1 =0 in H (cf. Shl99]). Obviously, (0.1) is a very particular case of this latter formula. In Section 8 we indicate how (0.1) may be used to yield approximate solutions to the Cauchy problem for Pu = f in D with data on . Write V0 for the orthogonal projection of H onto the null-space of A. Consider the operator

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R = P1=0 (I ; A) in H with a domain Dom R consisting of those f 2 H for which this series converges. If f 2 Dom R, then A Rf = f ; V f . If moreover f is orthogonal to the null-space of A, then A Rf = f . Conversely, if f = Au, for some u 2 H , then f 2 Dom R and f ? ker A. On the other hand, the equation Bu = f , for a non-necessarily selfadjoint operator B 2 L(H H~ ), reduces to B Bu = B f under the additional condition f ? ker B . Assuming = 6 , we apply this approach to the operator B = P whose adjoint is Pv. In Section 9 we derive in a similar way an interesting formula for solutions of the generalised Zaremba problem in D, the Dirichlet data being given on = 6 . 0

Finally, Section 10 deals with some examples which concern the case where is an entire component of the boundary surface. The advantage of this situation is that we need not invoke any weighted Sobolev spaces. Using explicit formulas for the Green function of D0 n we are able to construct approximate solutions for the Cauchy and Zaremba problems in D with data on , thus recovering the results of NS96].

1 Selfadjoint operators Let H be a Hilbert space with a scalar product (: :)H and A be a bounded selfadjoint operator in H . We also assume that A is non-negative, i.e., (Au u)H 0 for every u 2 H . It is well known that if H is a complex Hilbert space then any non-negative operator is selfadjoint. We can certainly assume that kAk 1, since otherwise we replace A by its multiple A=kAk.

Problem 1.1 Let f 2 H be given. Find (if possible) an element u 2 H

such that Au = f .

Generally speaking, Problem 1.1 is ill-posed, i.e., it may happen that the image of A is not closed. This means that no solution exists for some data f 2 H and the solution, if exists, do not depend continuously on the data. Hence it follows that the solvability conditions for the problem can not be described in terms of continuous linear functionals. Let IdH stand for the identity operator in H . If it causes no confusion we write simply Id instead of IdH . As usual we denote ker A the null-space of A. If V is a closed subspace of H , we write V for the orthogonal projection of H onto V .


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Theorem 1.2 In the strong operator topology of the space L(H ), the equal-

ities hold

lim AN = ker(I ;A) lim (I ; A)N = ker A : N !1 N !1

Proof. Since the operator A is continuous, its kernel ker(I ; A) is a closed

subspace of H . Therefore, when endowed with the Hermitian structure induced from H , it is a Hilbert space. Given any N = 1 2 : : :, the spectral theorem for bounded selfadjoint operators yields

AN =

Z1+0

N dI ()

(1.1)

;0 where (I ())01 is a resolution of the identity in H , corresponding to the selfadjoint operator 0 A I (see, for instance, Sections 5, 6 in Yos65, Ch. XI]). Passing to the limit in (1.1) one obtains lim AN = I (1 + 0) ; I (1 ; 0): N !1

Since I () is a spectral function, the operator I (1+0);I (1;0) is an orthogonal projection of H onto a closed subspace V (1) in H . Obviously, N (I ; A) Nlim !0 A u = 0 for every u 2 H , i.e., V (1) ker(I ; A). Finally, if u 2 ker (I ; A) then u = Au + (I ; A) u = Au = AN u for every N 0. Therefore, N u = Nlim !1 A u and so V (1) = ker(I ; A) whence lim AN = ker(I ;A): N !1

In order to complete the proof it suces to observe that since A is a selfadjoint non-negative operator with kAk 1, the operator I ; A has the same properties.

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Corollary 1.3 In the strong operator topology of the space L(H ), the equal-

ities hold

1 N + P A (I ; A) I = Nlim A !1 =0 1 N+P I = Nlim ( I ; A ) (I ; A) A: !1

(1.2)

=0

Proof. The equality A + (I ; A) = I implies

X

N ;1

A (I ; A) =0 N ;1 X = (I ; A)N + (I ; A) A

I =

AN

+

=0

for every 2 N. Using Theorem 1.2 we can pass to the limit, when N ! 1, thus obtaining (1.2). We use Corollary 1.3 in order to obtain a solvability condition for Problem 1.1. Theorem 1.4 Problem 1.1 is solvable if and only if the Neumann series 1 X u = (I ; A) f =0

converges in H . Moreover, if Problem 1.1 is solvable then Au = f . Proof. Let Problem 1.1 have a solution u~ 2 H . Then Corollary 1.3 implies that the series 1 1 X X (I ; A) Au~ = (I ; A) f =0 =0 = u converges in H . Conversely, let the series u converge in H . Since the operator I ; A is continuous it follows that u ; Au = (I ; A) u 1 X = (I ; A) f =1

= u ; f: Hence Au = f and Problem 1.1 is solvable.

Theorem 1.2 implies that the solution u of Theorem 1.4 is a unique solution to Problem 1.1 orthogonal to ker A.


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Remark 1.5 Clearly, the elements u(N ) =

X

N ;1 =0

(I ; A) f

can be regarded as approximate solutions to Problem 1.1, even if no real solution exists.

2 Bounded operators

Consider now a more general situation. Let H , H~ be Hilbert spaces, and B 2 L(H H~ ) a bounded operator. Without loss of generality we again restrict our attention to those B which satisfy kB k 1.

Problem 2.1 Let f

Bu = f .

Let B : H~ spaces.

2 H~ be given.

Find an element u

2 H such that

! H be the adjoint of B : H ! H~ in the sense of Hilbert

Lemma 2.2 Problem 2.1 is solvable if and only if

1) there is a u 2 H such that B B u = B f 2) (f g)H~ = 0 for all g 2 ker B .

Proof. Indeed, from Bu = f it follows readily that B Bu = B f . Con-

versely, if

B Bu = B f f ? ker B then B (Bu ; f ) = 0 whence

(Bu ; f Bu ; f )H~ = (Bu Bu ; f )H~ ; (f Bu ; f )H~ = 0

i.e., Bu = f . It is easy to see that A = B B is a bounded non-negative self-adjoint operator in H satisfying kAk 1. Therefore Problem 2.1 is equivalent to Problem 1.1, with f replaced by B f , f being orthogonal to the null-space of B .

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Theorem 2.3 In the strong operator topology of the space L(H~ ), the equal-

ity holds

(ker B ) =

1 X

?

=0

B (I ; B B ) B :

Proof. It is easy to see that

(IdH~ ; BB ) B = B (IdH ; B B )

whence

(IdH~ ; BB ) BB = B (IdH ; B B ) B

for all 0. Note that BB is a bounded selfadjoint non-negative operator in H~ satisfying kBB k 1. According to Theorem 1.2 and Corollary 1.3, we get 1 X (ker B ) = (IdH~ ; BB ) BB =0 1 X = B (IdH ; B B ) B ?

=0

as desired.

Corollary 2.4 Problem 2.1 is solvable if and only if

P (I ; B B ) B f converges in H 1) the series u = 1 =0 2) (f g)H~ = 0 for all g 2 ker B . Moreover, if Problem 2.1 is solvable then the series u is one of its solutions. Proof. This follows immediately from Lemma 2.2 and Theorems 1.4 and

2.3.

In the sequel we will discuss some applications of this approach to elliptic equations (cf. also Shl99]).

3 Green function Fix an interval = (a b) in the real axis. Consider the following Dirichlet problem in the plane with a crack along . Given any u0 2 Cloc(), nd a harmonic function u in R2 n , such that u extends continuously to from both the upper and lower half-planes and u = u0 on .


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We look for a solution of this problem by the layer potentials method. To this end, denote by G(x) = 21 log jxj the standard fundamental solution of convolution type to the Laplace operator in the plane. Assuming u to have at a and b poles of at most nite order, we conclude that u extends to a distribution on R2 whence

Z

u(x) = ue(x) + ((@=@y2)G(x ; y) U0(y) ; G(x ; y) U1(y)) dy1

(3.1)

for x 62 , where ue is a harmonic function on all of R2, and U0 U1 2 E 0(R) are supported on . When crossing the interval , the double layer potential has the jump U0(x0) at each Lebesgue point x0 2 of U0. Since we require u to be continuous up to , it follows that U0 should vanish on . Hence U0 is a nite linear combination of the derivatives of the Dirac delta-functions supported at a and b. Substituting this into (3.1) gives

u(x) = ue(x) +

J (a) X

@ j 1

cj (a) @x 1 j =0

Z

@ j 1 ;x2 J (b) ;x2 + X 2 (x1 ; a)2 + x22 j=0 cj (b) @x1 2 (x1 ; b)2 + x22

; G(x ; y) U1(y) dy1 for x 62 . The second term on the right hand side of this formula is a harmonic function in R2 n fag, vanishing both on and at the point at in nity. The

third term bears the same properties, with a replaced by b. Hence, to achieve the uniqueness in our problem, we need to impose additional conditions on the behaviour of u near the boundary of . They can be formulated in terms of belonging of u to weighted Sobolev spaces near the singular points, namely u 2 H s (B n ), 0. Here, B is any disk in the plane containing , and s (B n ) with nite H s (B n ), for s 2 Z+, is the space of all functions u 2 Hloc norm

11=2 0Z X min(jx ; aj jx ; bj)2(jj;)jDu(x)j2dxA : kukH (Bn E) = @ s

B n jjs

Further, the function ue is not uniquely determined by the data of the problem, too. Indeed, ue(x) = x2 is harmonic on all of R2 and vanishes on . To specify ue we impose on u the condition that u(x) has a limit as jxj ! 1.

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this limit by c, we see at once ue(x) c, for the limit of the integral RDenoting G(x ; ) U dy , as jxj ! 1, coincides with that of a constant multiple of 1

1

G(x). To show this, observe that the dierence G(x ; y) ; G(x) tends to 0, when jxj ! 1, the limit being uniform in y 2 . The simple layer potential of a continuous density on is a continuous function away from the endpoints of in R2. From what has already been proved it follows that a function u of the form (3.1) is a solution to our crack problem if and only if ue(x) c, U0(x) 0 and the density U1 satis es the equation Zb 1 u0(t) = c ; 2 log jt ; sj U1(s) ds t 2 (a b): (3.2)

a

Note that (3.2) is a Fredholm equation of the rst kind, whose kernel has a weak singularity on the diagonal. It was Carleman C22] who rst solved the equation (3.2) in a closed form. Assume that u0 is a C 1 function on (a b) and the derivative of u0 is of the form f (t)=(t ; a)1;"(b ; t)1;" , with " > 0, f satisfying a H"older condition on (a b). We look for a solution to (3.2) of the form U (s)=(s ; a)1;"(b ; s)1;" , where " > 0 and U meets a H"older condition on (a b). Then, the result of Carleman C22] reads as follows.

Lemma 3.1

1) If b ; a 6= 4, then, for each u0, the equation (3.2) has a unique solution given by

U1(s) = 2 p

1 Z b p(s ; a)(b ; s) Z b c ; u0(t) ! ( t ; a )( b ; t ) 1 a s ;t u00(t) dt + log b;a a p(t ;a)(b; t) dt : 4

2) If b ; a = 4, Rthen in order that (3.2) be solvable it is necessary and su cient that ab pc(;t;ua0)((tb);t) dt = 0. Under this latter condition, we have

U1(s) = 2 p

1 (s ; a)(b ; s)

Z b p a

(t ; a)(b ; t) u0 (t) dt + C 0 s;t

where C is an arbitrary constant.

Proof. See ibid. and Section 55 in Gakhov G77].

!


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By the very construction of reducing (3.2) to a singular integral equation with the Cauchy kernel on , the solution is obtained in the form U1(s) = U 0(s), with U (a) = U (b) = 0 (cf. (55.24) in G77]). Hence it follows that the integral

Z

G(x ; y) U1 (y) dy1 = G(x ;

j

(y1 0))U (y1) yy11 ==ba +

Z

(@=@x1) G(x ; y) U (y) dy1

tends to zero, when jxj ! 1. Thus, if b ; a 6= 4, then the unique solution to the Dirichlet problem in R2 n with the data u0 on and c at the point at in nity is given by

Z

u(x) = c ; G(x ; y) U1(y) dy1 x 2 R2 n

(3.3)

the density U1 being de ned in Lemma 3.1. As explained above, this solution is the best of many other possible solutions. Using the explicit solution of the Dirichlet problem in R2 n , we can construct the Green function of this domain. To this end, x any point y 2 R2 n and denote by R( y) the unique harmonic function in R2 n , such that R(x y) = G(x ; y) on and R(x y) ! 0 as jxj ! 1. From (3.3) we deduce 1 (R2 n ). that R(x y) is a C 1 function of the parameter y with values in Cloc The dierence G (x y) = G(x ; y) ; R(x y) is the desired Green function of R2 n . For x 6= y, G (x y) is C 1 up to from both sides of this interval, hence the only singularities of G (x y) away from the diagonal lie at the corner points @. In fact, we have 2 Z b (t y) dt Zb @ G(x y) = G(x ; y) + a G(x ; s) @s Z (s) a Z (t) t ; s ds (3.4) for x y 2 R2 n , where any point s 2 (a b) is identi ed with the point (s 0) 2 R2 and p Z (s) = (s ; a)(b ; s) Z b G(s ; y) 1 log( b ; t ) (t y) = G(t ; y) ; ds: log b;4a a Z (s) By (3.4), it is possible to get an explicit estimate of G (x y) near the corner points a and b. However, for our purposes it will be sucient to know that G (x y) belongs to the space H 10(B n ) with respect to either of the variables x and y while the other variable lies away from B , B being a disk containing . Remark 3.2 The Green function G (x y) allows one to construct the solution to the Dirichlet problem in R2 n by Z @ Z @ + + ; ; u(x) = c + G(x ) (u ; c) ds ; @y2 G(x ) (u ; c) ds @y2

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where u are the prescribed limit values of u on from the upper and lower half-planes, respectively, and c is the prescribed limit of u(x) when jxj ! 1.

4 A crack problem

Let X be an open set in Rn and P 2 Di p(X # E F ) an elliptic dierential operator of order p on X , where E = X C k , F = X C l . Having xed Hermitian metrics on E and F , denote by P 2 Di p(X # F E ) the formal adjoint for P . Then, = P P is an elliptic dierential operator of order 2p and type E ! E . Assume that bears the uniqueness property for the Cauchy problem in the small on X . Then it has a two-sided fundamental solution G on X . Let D be a relatively compact domain in X . We emphasise that the boundary of D need not be smooth. Denote by SP (D) the space of all solutions of the system Pu = 0 in D. In Section 5 we construct a special scalar product h( ) on the weighted Sobolev space H pp(D E ). This is obtained by using a fundamental solution of having special properties at the boundary of a larger domain O b X with singular boundary. Throughout this section we will assume that the boundary of D consists of two smooth pieces, namely @ D = (@ D n ) where is an open connected subset of @ D with smooth boundary. Fix a domain D0 with C 1 boundary, such that D b D0 b X . ;1 of order (p ; 1) on the smooth Let we be given a Dirichlet system fBj gpj=0 part of @ D @ D0, with Bj being of type E ! Fj and order mj . Denote by fCj gpj=0;1 the Dirichlet system adjoint to fBj g with respect to the Green formula for P (cf. Tar95, 9.2.1]). As is well-known, Cj is of type F ! Fj and order p ; 1 ; mj . The only singularities of the coecients of fCj g lie on the interface @ @ D. By the above, there is a Green operator GP for P whose restriction to the smooth part of @ D @ D0 is GP (F g u) = (t(u) n(g))y ds, where ds is the area form of @ (D0 n D) and

t(u) = n(g) =

L B u j j =0 pL ;1 ;1 F Cj F g j =0 p;1

j

for u 2 C 1(E ) and g 2 C 1(F ). Moreover, F and F are sesquilinear bundle isomorphisms F ! F and Fj ! Fj induced by the Hermitian metrics on F and Fj , respectively. j


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Recall that H s (D E ) stands for the weighted Sobolev space of sections of E over D of smoothness s and weight , both s and being real numbers. For s 2 Z+, it coincides with the completion of the C 1 sections of E vanishing near @, with respect to the norm

0Z 11=2 X kukH (DE) = @ dist(x @)2(jj;)jDu(x)j2dxA : s

D jjs

We denote by H s; 12 ; 21 (@ D E ) the space consisting of the traces of functions u 2 H s (D E ) on the smooth part of @ D. As usual, we endow this space by the quotient norm# an inner description is available, too. In his edge calculus, Schulze Sch98] uses another scale of weighted Sobolev spaces, denoted by W s (D E ). While H s (D E ) and W s (D E ) are quite dierent from each other, for arbitrary s and , they coincide on the diagonal s = , i.e., H ss (D E ) = W ss(D E ) for all s 2 Z+ (cf. Proposition 3.1.5 in Schulze Sch99]). This is just the case in the present paper where we deal, for the most part, with the spaces H pp(D E ).

Lemma 4.1 Let 2 R. For each u 2 H p (D E ) and g 2 H pp; (D F ),

we have

Z

Z

((Pu g)x ; (u P g)x) dx: D Proof. Indeed, if g = g vanishes near @, then the equality is simply a Green formula for P (cf. Tar95, 9.2.1]). Since @D

(t(u) n(g))x ds =

Z (t(u) n(g ))x ds c kukH @D

and

1

p; ;

Z (Pu g )x dx c kukH Z D (u P g )x dx c kukH

1 2

p

(@ DE )

kg kH

kg kH (DE ) kg kH

(DE )

1

1 2

0p;

(DF )

p; p; ;

(@ DF )

0 (DF ) D with c a constant independent of u and g , the proof is completed by a passage to the limit in the formula for g vanishing near @ and approximating an arbitrary g 2 H pp; (D F ).

pp;

Consider the following Dirichlet problem for the Laplacian = P P in the domain O = D0 n having a crack along . Let s 2 Z+ satisfy s p, and

18


let 2 R. Given any u0 2 H s;m ; 21 ;m ; 12 (@ O Fj ), nd a u 2 H s (O E ) such that

u = 0 in O (4.1) t(u) = u0 on @ O: ;1 is a Dirichlet system of order (p ; 1), we may solve (4.1) Since fBj gpj=0

rst in the class of weighted Sobolev spaces, thus reducing the problem to that of nding a u~ 2 H s (O E ) satisfying ~u = f in O and t(~u) = 0 on @ O, for given f 2 H s;2p;2p(O E ). The advantage of using this reduction lies in the fact that the latter problem can be posed as a variational one. Namely, given any f 2 H s;2p;2p(O E ), p, nd a u~ 2 H s (O E ) such that t(~u) = 0 on @ O and Z Z (P u~ Pv)x dx = (f v)x dx (4.2) O O for all v 2 H s (O E ) satisfying t(v) = 0 on @ O. Note that the restriction p is necessary because otherwise the pairings in (4.2) are not de ned. If s = = p, then (4.2) is uniquely solvable for all f 2 H ;p;p (O E ). Indeed, denote by H the subspace of H pp(O E ) consisting of all u satisfying R t(u) = 0 on @ O. The sesqui-linear form (u v)H = O (Pu Pv)xpdx is easily seen to de ne a scalar product on H . Moreover, the norm u 7! (u u)H on H is equivalent to that induced by H pp(RO E ), hence H is Hilbert. For every f 2 H ;p;p (O E ), the functional v 7! O (f v)x dx is continuous on H . By the Riesz theorem, this functional can be written in the form (4.2) with some u 2 H , as desired. That we have chosen s = p is not important at all, for the Dirichlet problem meets the Lopatinskii condition on the smooth part of @ O. However, the exponent = p is of exceptional character. More precisely, the following is true. j

j

Lemma 4.2 There is a closed set R with the property that, if 62 , then the Dirichlet problem (4.1) has a unique solution u 2 H s (O E ), for each 1 1 data u0 2 H s;m ; 2 ;m ; 2 (@ O Fj ). Moreover, p 62 . j

j

Proof. Indeed, (4.1) is an elliptic boundary value problem in the domain D0

with a crack along the smooth surface . It can be treated in the framework of analysis on manifolds with boundary and edges of codimension 1 on the boundary (cf. Nazarov and Plamenevskii NP91], Schulze Sch92], Duduchava and Wendland DW95], Rabinovich, Schulze and Tarkhanov RST98], etc.). Either of these calculi is applicable. The fact that p 62 is a consequence of the formal selfadjointness of the Dirichlet problem (see Sections 6.1.3 and 8.4.2 in NP91]).


19

In particular, we are able to invoke the classical scheme of constructing the Green function of an elliptic boundary value problem by improving a two-sided fundamental solution, the domain being O = D0 n .

Proposition 4.3 Suppose 1 62 . Then, there exists a Green function G(x y) of problem (4.1), i.e., a two-sided fundamental solution of in O, such that

1)

G extends to a smooth matrix-valued function away from the diagonal 0 in (D n @ ) (D0 n @), the smoothness near being understood sidewise

2) t (G ( y)) = 0 on @ D0 , for each y 2 O, the operator t acting in the variable x.

Proof. It is easy to verify that, given any xed y 2 O, the function u0 = t (G( y)) belongs to H 1;m ; 12 (@ D0 Fj Ey) for all < 1. Denote by R( y) the unique solution of problem (4.1) corresponding to this u0. Then G (x y) = G(x y) ; R(x y) bears all the desired properties. j

If multiplied by an excision function of y, the Green function G ( y) actually belongs to H 1p(O E Ey), for each y 2 O, as is seen from the proof of Proposition 4.3. For more details we refer the reader to Section 6 of NP91, Ch. 8]. When compared to the explicit formula (3.4), the proof highlights the structure of G . Indeed, G can be constructed within the pseudodierential calculus on O developed in Schulze Sch98], where O is thought of as a manifold with edge @ on the boundary. Thus, G inherits the structure of operators in this calculus, cf. Section 3.4 ibid..

Remark 4.4 Since the problem (4.1) is formally selfadjoint, the Green

function G actually satises G (x y ) = G (y x) on not use this symmetry.

O O.

5 Adjoint operator

However, we will

Let S (D^0 n D ) be the set of all solutions to the equation u = 0 in D0 n D , such that u has nite order of growth near @ D0 and t(u) = 0 on @ D0. Using Lemma 4.2 for D0 n D instead of D0 n , we obtain readily a linear isomorphism t+ H p (D0 n D E ) \ S (D^0 n D ) ;! H p;m ; 12 ;m ; 12 (@ D Fj ) j

j

20


given by u operator

7! t(u) j@D. Finally, composing the inverse t;+1 with the trace t H p (D E ) ;! H p;m ; ;m ; (@D Fj ) ;

1 2

j

j

1 2

yields a continuous linear mapping E H p (D0 n D E ) \ S (D^0 n D ): H p (D E ) ;!

(5.1) Suppose p. Then we have Pu 2 L2(D F ) and P E (u) 2 L2(D0 n D E ) for each u 2 H p (D E ). Consider the Hermitian form

h(u v) =

Z

D

(Pu Pv)x dx +

de ned for u v 2 H p (D E ).

Z

D nD

0

(P E (u) P E (v))x dx

Proposition 5.1 The Hermitian form h( ) denes a scalar product in

H p (D E ).

0 in D. To do this, pick a u 2 D E) with h(u u) = 0. Then, u 2 SP (D) and E (u) 2 SP (D0 n D ). By the construction of E , we have t(u) = t(E (u)) on the smooth part of the boundary of D. According to Theorem 1.3.3 in Tar97] there is a solution u~ 2 SP (D0 n @) such that u~ = u in D and u~ = E (u) in D0 n cD . In particular, t(~u) = 0 on @ D0. Since the uniqueness property holds true for the Cauchy problem in the small for the Laplacian P P , it does so for P . This gives u~ 0 on all of D0 n @ (cf., for instance, Tar95, 10.3.1]). In particular, u = 0 in D, as desired. Proof. It suces to prove that h(u u) = 0 implies u H p (

Set

(x y) = F P (y D) ;E1 G (x y) for (x y) 2 (D0 n ) (D0 n ). Moreover, introduce the double layer and volume potentials by ;Pdlu(x) = @RD (t(u) n(;1(x )))y ds x 2 D0 n @D# Pvf (x) = DR (f ;1(x ))y dy x 2 D0 for u 2 H p (D E ) and f 2 H 0;p(D F ). By the above, the integrals make sense provided p. Since Pdlu does depend only on the Cauchy data t(u1) of u 1on @ D, the designation Pdlu0 still makes sense for any u0 2 H p;m ; 2 p;m ; 2 (@ D Fj ). Let (Pvf ); and (Pvf )+ be the restrictions of Pvf to D and D0 n D , respectively. j

j


21

Lemma 5.2 There exists a constant c > 0 with the property that, for every

f 2 H 0;p(D F ), we have

k(Pv+f );kH (DE) c kf kH k(Pvf ) kH (D nD E) c kf kH p

p

0 ;p

0

0 ;p

(DF ) (DF ) :

Proof. This follows from the boundedness of edge pseudodierential operators in the weighted Sobolev spaces, Sch98]. This lemma gives us a hint that the proper choice of the weight exponent should be = p.

Z

Lemma 5.3 For every u 2 H p (D E ),

Z

(Pu ;1(x ))y dy + (P E (u) ;1 (x ))y dy = D D nD

0

u(x)

x 2 D E (u)(x) x 2 D0 n D :

Proof. Since G (x y) diers from G(x y) by a kernel R(x y) which satis es 0 (y D)R(x y) = 0 over O O, it follows that P = G P P = (G ; R) = Id

on compactly supported sections of E jO . In other words, is a left fundamental solution of P on O. By the Green formula,

;

Z

;t(E (u)) n(;1(x ) ds y @ (D nD )

0 Z ; ; 1 + P E (u) (x ) dy =

x 2 D x ) x 2 D0 n D : E ( u )( D nD

We now make use of the fact that t(E (u)) = t(u) on @ D and t(E (u)) = 0 on @ D0. Hence we deduce that the integral over @ (D0 n D ) on the left hand side is equal to ;Pdlu(x), for all x 2 D0 n @ D. This latter potential can in turn be expressed from the Green formula for u over the domain D, thus implying the 0

0

y

desired equality.

If 1, then the elements of H p (D E ) vanish on the boundary of , to which refers the weight exponent . Denote by H p ] (D E ) the closure in p 1 H (D E ) of the C sections of E vanishing near .

22


Proposition 5.4 Suppose u p2p;

each v 2 H ]

2 H p (D E ) and f 2 H 0;p(D F ).

(D E ), it follows that

For

R

h (Pvf v) = (f Pv)xdx DR h (Pdlu v) = (P E (u) P E (v))x dx: D nD

Proof. Indeed, if f 2 H p;p(D F ) and v 2 H p2p; (D E ), then integration by parts gives 0

Z

D

(f Pv)xdx ;

Z

D

(P f v)xdx =

Analogously,

Z D nD 0

(P E (u) P E (v))x dx = ;

Z @D

Z

(n(f ) t(v))xds:

(5.2)

(n(P E (u)) t(v))xds

(5.3)

@D

for all u 2 H 2p (D E ) and v 2 H p2p; (D E ), because t(E (v)) = t(v) on @ D and t(E (v)) = 0 on @ D0. Fix u 2 H 2p (D E ) and v 2 H p2p; (D E ), and apply formula (5.2) for f = Pu. Using (5.2) and (5.3), we get

h(u v) =

Z

(n(Pu) ; n(P E (u)) t(v))x ds +

Z

(u v)x dx: D 1 (D F ). In this case integrating by parts shows readily Let us take f 2 Ccomp that Z ; Pvf (x) = D P f ;E1G(x ) y dy for all x 62 @ D. From Proposition 4.3 we deduce that Pvf is a C 1 section of E over D0 n , such that t (Pvf ) vanishes on both and @ D0. Moreover, it satis es Pvf = 0 on D0 n D . Hence it follows that E (Pvf ); is simply equal to the restriction of Pvf to D0 n D , for t (Pvf )+ = t (Pvf ); on @ D n . Now we can substitute (Pvf ); for u in the formula above, to obtain @D

Z ; Z ; + h (Pvf v) = n(P (Pvf ) ) ; n(P (Pvf ) ) t(v) x ds + @D

D

(Pvf v)x dx:

Since Pvf is smooth away from in D0, the jump of n(P Pvf ) across @ Dn is zero. On the other hand, this jump has nothing to do on the rest part of the boundary of D, provided that t(v) = 0 on . Thus, the rst summand on the right hand side of the last equality is equal to zero, if v varies over


23

H p]2p; (D E ). Taking into account that Pvf (x) = P f (x) for all x 2 D, we get Z h (Pvf v) = (P f v)x dx ZD = (f Pv)x dx: D 1 (D F ) is dense in H 0;p(D F ), this formula holds for every Since Ccomp f 2 H 0;p(D F ) and v 2 H p]2p; (D E ), as desired. Finally, we can express the double layer potential Pdlu from the Green formula for P in D as Pdlu = u ; PvPu. Hence h (Pdlu v) = hZ(u v) ; h (PvPu v) = (P E (u) P E (v))xdx D nD

for any u 2 H p (D u) and v 2 H p]2p; (D E ), as is clear from what has already been proved. This establishes the formula. 0

Not only does Proposition 5.4 specify the adjoint operator for P under the scalar product h(u v), but does also show the self-adjointness of the double layer potential. Lemma 5.5 The mappings t;, t+ induce topological isomorphisms of Hilbert spaces, namely t H p (D E ) \ S (D) ;! H p;m ; 21 ;m ; 12 (@ D Fj ) t+ H p (D0 n D E ) \ S (D^0 n D ) ;! H p;m ; 12 ;m ; 12 (@ D Fj ): Proof. This follows from Lemma 4.2 as is explained at the very beginning of this section. ;

j

j

j

j

Proposition 5.6 The topologies induced in H pp(D E ) by h( ) and by

the original scalar product are equivalent.

Proof. As the mapping (5.1) is continuous, we see, for any u 2 H pp(D E ),

that

h(u u) = kPuk2L2(DF ) + kP E (u)k2L2 (D nD F )

2 2 c kukH (DE) + kE (u)kH (D nD E) c (1 + c0) kuk2H (DE) 0

pp

pp

pp

0

24


the constants c and c0 being independent of u. Here, we use the identi cation L2(D F ) = H 00(D F ) and the fact that P acts continuously from H s (D E ) to H s;p;p (D F ). Conversely, Lemmas 5.2 and 5.3 imply that 1 kuk2 2 2 2 H (DE) kPv PukH (DE) + kPv P E (u)kH (DE) c kPuk2L2(DF ) + c kP E (u)k2L2(D nD F ) = c h(u u) with c a constant independent of u. This completes the proof. pp

pp

pp

0

As mentioned above, Proposition 5.4 allows one to specify the adjoint operator of P with respect to the scalar product h(u v) on H pp ] (D E ) as the volume potential Pv. Theorem 5.7 In the Hilbert space H pp] (D E ) endowed with the scalar product h( ), we have kP k 1 and P = Pv . Proof. This is an immediate consequence of Propositions 5.1, 5.4 and 5.6.

6 The case of constant coecients Note that we may use similar arguments for the case where P is a rst order elliptic homogeneous dierential operator with constant coecients in X = Rn, n 2, and D0 is taken to be the whole X . To this end, however, we need to correct the scalar product h( ). As = P P is a second order elliptic homogeneous dierential operator with constant coecients, it has a fundamental solution of convolution type x;y G(x ; y) = G1 jx ; yj jx ; yj2;n + G2 (x ; y) log jx ; yj where G1(z) is a (k k)-matrix of real analytic functions near the unit sphere in Rn, and G2(z) is a (k k)-matrix of homogeneous polynomials of degree 2 ; n. In particular, G2(z) 0 for n > 2. For example, if P is the gradient operator in Rn, then P P is the ;1 multiple of the Laplace operator in Rn. If P is the Cauchy-Riemann system in C N ( = R2N), then P P is the ; 14 multiple of the Laplace operator in R2N . These examples extend to various matrix factorisations of the Laplace operator in Rn. Let K be a compact set in Rn. Denote by S (R^ nnK ) the set of all functions u on Rn n K with values in C k , such that


25

1) u = 0 in Rn n K # 2) there exists a limit limjxj!1 u(x) = 0, for n > 2, or limjxj!1 u(x) 2 C k , for n = 2. Let moreover SP (R^ n n K ) be the closed subspace of S (R^ n n K ) consisting of those functions SP (Rn n K ) which satisfy limjxj!1 u(x) = 0. Fix a point x0 in the domain D. Then Theorem 3.2.15 from Tar95] implies that every function u 2 S (R^ n n K ) can be expanded into a Laurent series

u(x) = c(u x0) +

X

2Z+ n

D G(x ; x0) c(u x0)

in the complement of a closed ball B with centre x0 and a radius R depending on P and K . The series converges absolutely and uniformly in Rn n B , and the coecients c(u x0) and c(u x0) are uniquely de ned. In fact, c(u x0) = 0, for n > 2, and c0(u x0) = 0, for n = 2. Mention that the conditions at the point at in nity imposed on the elements of S (R^ n n K ) are necessary for the well-posedness of the Dirichlet problem for in a domain with compact complement. Thus, instead of Lemma 4.2 we need the following one. Lemma 6.1 Let n > 2 and be close to 1. For each u0 2 H s; 12 ; 12 ( E ), there exists a unique function u 2 H s (Rn n E ) \S (R^ n n ) satisfying u = u0 on .

Proof. The proof is just in the same framework as that of Lemma 4.2 because the ellipticity at the point at in nity is ful lled. The lemma is still true for n = 2 if we slightly modify the formulation by requiring u 2 H s (B n E ), for each ball B containing . Since the Dirichlet problem for in Rn n is elliptic, the solution u can be represented by

Z; u(x) = u n(P ;E1 G (x ))+ ; n(P ;E1 G (x )); y ds

where G (x y) is what is usually called the Green function of the problem and n(P ;1 G (x )) are the limit values of n(P ;1 G (x )) on from Rn n D and D, respectively.

Proposition 6.2 There is a Green function G (x y) of the Dirichlet prob-

lem for in such that

Rn

n , i.e., a two-sided fundamental solution of in Rn n ,

26 1)


G extends to a smooth matrix-valued function away from the diagonal n in (R n @) (Rn n @), the smoothness near being understood

sidewise 2) G ( y) vanishes on and the di erence G( y) ; G ( y) behaves at the point at innity as an element of S (R^ n n ), for each y 2 Rn n . Proof. In view of Lemma 6.1, this is a very particular case of Proposition 4.3.

Solving the Dirichlet problem for in Rn n D instead of Rn n , we get a linear isomorphism t+ H 1 (B n D E ) \ S (R^ n n D ) ;! H 12 ; 21 (@ D E ) by u 7! u j@D, where B is any ball containing D Then, composing the inverse t;+1 with the trace operator t H 1 (D E ) ;! H 21 ; 12 (@ D E ) we arrive at a continuous linear mapping E H 1 (B n D E ) \ S (R^ n n D ) H 1 (D E ) ;! (cf. (5.1)). Theorem 6.3 The Hermitian form ;

Z

Z

h(u v) = (Pu Pv)y dy + (P E (u) P E (v))y dy + (E (u)(1) E (v)(1)) D R nD is a scalar product on the space H 11(D E ) inducing the topology equivalent to n

the original one. Proof. This follows by the same method as in the proof Proposition 5.6. The main ingredient thereof is Lemma 5.3 which still remains valid in the case D0 = Rn.

For u 2 H 1 (D E ), we introduce the double layer potential Pdlu and, for f 2 H 0;1(D F ), the volume potential Pvf on Rn n @ D by the same formulas as in Section 5. Proposition 6.4 Suppose u 2 H 1 (D E ) and f 2 H 0;1(D F ). For ; 1 2 each v 2 H ] (D E ), it follows that R h (Pvf v) = (f Pv)xdx DR h (Pdlu v) = R nD (P E (u) P E (v))x dx + (E (u)(1) E (v)(1)) : Proof. The proof is similar to the proof of Proposition 5.4. n


27

7 Iterations The following theorem can actually be formulated in the abstract context of selfadjoint operators in Hilbert spaces, cf. Theorem 1.2.

Theorem 7.1 In the strong operator topology on the space L(H pp] (D E )),

we have

P

lim N N !1 dl lim ( P )N N !1 v

= V1 = V0

P where 0 V0 = fu 2 H pp ] (D E ) : P E (u) = 0 in D n Dg V1 = fu 2 H pp ] (D E ) : Pu = 0 in Dg: Proof. First, Theorem 5.7 implies that the integrals Pdl and PvP de ne bounded selfadjoint non-negative operators in the Hilbert space H pp ] (D E ) with the scalar product h( ). Moreover, their norms do not exceed 1.

By the spectral theorem for bounded selfadjoint operators we conclude readily that Z 1+0 N Pdl = 0; N dI () (7.1) where (I ())01 is a resolution of the identity in the Hilbert space H pp ] (D E ) corresponding to the selfadjoint operator 0 Pdl I (see, for instance, Sections 5, 6 in Yos65, Ch. XI]). Passing to the limit in (7.1) yields

P

lim N N !1 dl

= I (1 + 0) ; I (1 ; 0)

the operator on the right side being an orthogonal projection of H pp ] (D E ) onto a (closed) subspace V (1). Obviously, N (I ; Pdl) Nlim !0 Pdl u = 0

for all u 2 H pp ] (D E ), i.e., V (1) ker(I ; Pdl). Finally, if u 2 ker (I ; Pdl) then

u = Pdlu + (I ; Pdl) u = Pdlu = PdlN u for every N 0. Hence it follows that N u = Nlim !1 Pdl u

28


and so V (1) = ker(I ; Pdl) whence lim P N = ker PvP : N !1 dl Arguing in the same way we obtain lim (P P )N = ker Pdl : N !1 v Now Proposition 5.4 implies that the null-space of PvP in H pp ] (D E ) copp incides with H ] (D E ) \ SP (D). Finally, Proposition 5.4 and Lemma 5.3, if combined with the Green formula for P , imply that Pdlu = 0 if and only if P E (u) = 0 in D0 n D . Recall that the operator P bears the uniqueness property for the Cauchy problem in the small on X . Hence, if the boundary of D is connected, Theorem 10.3.5 of Tar95] implies that V0 coincides with the space of all u 2 H pp(D E ) such that t(u) = 0 on @ D. On the other hand, if 6= , then the uniqueness theorem for P gives V1 0. Corollary 7.2 In the strong operator topology of the space L(H pp] (D E )), we have 1 X N I = Nlim (7.2) !1 Pdl + =0 P1dl PvP N + X (P P ) P : I = Nlim ( P P ) (7.3) v v dl !1 =0

Proof. Indeed, the Green formula Pdl + PvP = I in D implies I = =

PdlN + X Pdl PvP =0 N ;1 X N (PvP ) + (PvP ) Pdl N ;1

=0

for every N 2 N. Now using Theorem 7.1 we can pass to the limit for N ! 1, thus obtaining (7.3) and (7.2). For applications to the Cauchy problem for solutions of Pu = 0 in D with data on , we also need the following result. Corollary 7.3 Suppose 6= . Then, in the strong operator topology of L(L2(D F )), we have 1 X I = ker Pv + P Pdl Pv: =0


29

Proof. Arguing as in the proof of Corollary 7.2, with Theorem 7.1 replaced

by an abstract Theorem 2.2 of Shl99], we obtain 1 X N PP I = Nlim ( I ; P P ) + ( I ; P P ) v v v !1 =0 1 X = ker P Pv + (I ; P Pv) P Pv =0 on L2(

I being the identity operator D F ). 2 Assume that f 2 L (D F ) and P Pvf = 0 in D. As t(Pvf ) = 0 on , it follows, by the uniqueness property for the Cauchy problem in the small, that Pvf = 0 in D. Consequently, the kernel of P Pv on L2(D F ) coincides with that of Pv. To complete the proof, it suces to observe that (I ; P Pv ) P Pv = P (I ; PvP ) Pv = P Pdl Pv for any 2 Z+, which establishes the formula.

8 Cauchy problem Consider the following Cauchy problem, for the operator 1P and 1the Dirichlet ;1. Given f 2 L2(D F ) and u0 2 H p;m ; 2 p;m ; 2 ( Fj ), nd system fBj gpj=0 a section u 2 H pp(D E ) satisfying

Pu = f in D (8.1) t(u) = u0 on : This problem is well-known to be ill-posed unless = @ D. Using Corollary 7.2 we obtain easily approximate solutions to the problem. To this end, set Z; Pdlu0 (x) = ; u0 n(;1(x )) y ds x 2 D: (8.2) j

j

Note that the integral on the right hand side of formula (8.2) is well-de ned because of the properties of (x ) and u0. Theorem 8.1 The problem (8.1) has at most one solution. It is solvable if and only if f ? ker Pv and the series 1 X (8.3) u = Pdlu0 + Pdl Pv (f ; P Pdlu0) =0

30


converges in H pp(D E ). Moreover, the series, if converges, is the solution to this problem. Proof. For the uniqueness theorem for solutions of (8.1) we refer the reader to Tar95, 10.3.1]. Lemma 4.2 and Proposition 4.3 imply that the potential Pdlu0 belongs to H pp(D E ) \ S (D) and satis es t(Pdlu0) = u0 on . Thus, by setting u = Pdlu0 + u~ we reduce the problem (8.1) to the problem of nding a function u~ 2 H pp(D E ) such that

P u~ = f~ in D (8.4) t(~u) = 0 on where f~ = f ; P Pdlu0. It is a simple matter to see, by Theorem 5.7, that f~ ? ker Pv if and only if f ? ker Pv. Let the problem (8.4) be solvable. Then f~ ? ker Pv. Moreover, equality P 1 (7.2) implies that the series u~ = =0 Pdl Pvf~ converges and gives a solution to this problem. Therefore, the problem (8.1) is solvable, too, and its solution is presented by (8.3). Conversely, let f~ ? ker Pv and the series (8.3) converge in H pp(D E ). Then Corollary 7.3 implies that 1 X P u~ = P Pdl Pvf~ =0

f~ ; ker Pv f~ f~

= =

as desired. The theorem gains in interest if we realise that the null-space of Pv consists of all g 2 L2(D F ) satisfying P g = 0 in D and n(g) = 0 on @ D (cf. Lemma 10.2.20 in Tar95]). Hence, f ? ker Pv implies, in particular, that P1f = 0, for each dierential operator P1 over D satisfying P1P = 0.

9 Zaremba problem

Consider the following generalised Zaremba problem. Given f 2 H ;p;p (D E ) and u0 2 H p;m ; 12 p1 ;m ; 12 (1 Fj ) u1 2 H ;p+m + 2 ;p+m + 2 (@ D n Fj )

nd a section u 2 H pp(D E ) such that 8 u = f in D < t(u) = u0 on (9.1) : n(Pu ) = u1 on @ D n : j

j

j

j


31

It is worth pointing out that the trace of n(Pu) on @ Dn is not de ned for any u 2 H pp(D E ), for the order of n P is 2p ; 1. To cope with this, a familiar way is to assign an operator L with a dense domain Dom L ,! H pp(D E ) to (9.1), such that both t(u) and n(Pu) are well-de ned for all u 2 Dom L. More 1 (D n @ E ) with precisely, Dom L is de ned to be the completion of Ccomp respect to the graph norm of u 7! (u t(u) n(Pu)) in H pp(D E ) D N, where D = H p;m ; 12 p1 ;m ; 12 (1 Fj ) N = H ;p+m + 2 ;p+m + 2 (@ D n Fj ): For more details, see Roitberg Roi96] and elsewhere. Then, (9.1) de nes a continuous operator Dom L ! H ;p;p (D E ) D N by Lu = (u t(u) n(Pu)). If P is the gradient operator in Rn, then (9.1) is just the classical Zaremba problem in D. As the Dirichlet problem is elliptic, it is easy to reduce the mixed problem (9.1) to that with u0 = 0. To this end, set u = Pdlu0 + u~, the integral Pdlu0 being given by (8.2). Then the data in (9.1) transform to f~ = f and u~0 = 0, u~1 = u1 ; n (P Pdlu0). The advantage of studying the problem (9.1) with u0 = 0 lies in the fact that it can be thought of in the following variational sense. For any f and u1, as above, nd a u 2 H pp ] (D E ) such that j

j

j

Z

Z

j

Z

(Pu Pv)x dx = (f v)x dx + (u1 t(v))x ds (9.2) D D @D for all v 2 H pp ] (D E ). For Zaremba data f and u1, we introduce the simple layer and volume potentials by Pslu1 (x) = @DnR (u1 t(;1G(x )))y ds x 2 D# (9.3) G (D f ) (x) = DR (f ;1G(x ))y dy x 2 D where D is the characteristic function of D. Recall, cf. Remark 4.4, that t(;1G (x )) vanishes on , for each xed x 2 D0 n , hence the rst integral in (9.3) is actually over all of @ D.

Theorem 9.1 If =6 , then the problem (9.1) has no more than one

solution. It is solvable if and only if the series 1 X u = Pdlu0 + Pdl (G (D f ) + Psl (u1 ; n(P Pdlu0))) =0

(9.4)

converges in H pp (D E ). Moreover, the series, when converges, is the solution to this problem.

32


Proof. The uniqueness theorem for the problem (9.1) follows from the pp

variational setting (9.2) immediately. Indeed, let u 2 H ] (RD E ) be any solution of the problem corresponding to the zero data. Then D (Pu Pv)xdx = 0 for all v 2 H pp ] (D E ). In particular, substituting v = u we conclude that pp u 2 H ] (D E ) \SP (D). Since P bears the uniqueness property for the Cauchy problem in the small on X we see that u 0, which is our claim. Since t(;1G (x )) = 0 on , for each x 2 D, it follows from the Green formula for P that

Pv Pu = G (Du) + Psl n(Pu)

for all u 2 H pp(D E ). Hence, formula (7.2) shows that any solution u~ to the problem (9.1) with u~0 = 0 can be presented by the formula 1 X u~ = Pdl (G (D f ) + Psl u~1) : (9.5) =0

It follows that any solution to the general problem (9.1) can be written by formula (9.4). Conversely, let the series (9.4) converge to a function u 2 H pp(D E ). Then the series (9.5) converges, too, to the function u~ = u ; Pdlu0 lying in H pp ] (D E ). Since

Pv P u~

= (I ; Pdl) u~ 1 1 X X Pdl (G (Df ) + Psl u~1) ; Pdl (G (Df ) + Psl u~1) = =

we conclude that

=0

G (D f ) + Psl u~1

Z D

=1

(P u~ Pv)x dx = h (Pv P u~ v) = h (G (D f ) + Psl u~1 v)

for any v 2 H pp ] (D E ), the rst equality being due to Proposition 5.4. Set U = G (Df ) + Psl u~1. Using the fact that G (x y) is a two-sided fundamental solution for in D0 n , we get U ; = f E (U ; ) = U + where U ; and U + stand for the restrictions of U to D and D0 n D , respectively. Therefore, if v is a C 1 section of E vanishing in a neighbourhood of , we

Regularisation of Mixed Boundary Problems have

Z

D

(P u~ Pv)x dx = =

33

Z Z ; (f v)x dx + n(PU ; ) ; n(P E (U ; )) t(v) x ds ZD Z@D ; (f v)x dx +

n(PU ; ) ; n(PU + ) t(v) x ds:

D @D By a jump theorem for the simple layer potential (cf. Theorem 10.1.5 of Tar95]), the dierence n(PU ; ) ; n(PU +) is equal to u~1 on @ Dn . Therefore,

Z

Z

Z

(P u~ Pv)x dx = (f v)x dx + (~u1 t(v))x ds D D @D 1 (X E ) which gives (9.2) if combined with the fact that the functions v 2 Cloc pp vanishing near are dense in H ] (D E ). Thus, the problem (9.1) is solvable, which completes the proof. Were the problem (9.1) rst reduced to the particular case corresponding to f = 0 and u0 = 0, we would obtain yet another formula for approximate solutions of this problem. Theorem 9.1 just amounts to saying that the operator L bears the following two properties: 1) L is injective, and 2) (f u0 u1) 2 Ran L is equivalent to the convergence of the series (9.4) in H pp(D E ). Moreover, since the Zaremba problem is formally selfadjoint with respect to the Green formula for , it follows that the range of L is dense in H ;p;p (D E ) D N.

Remark 9.2 If (9.1) is Fredholm, the series (9.4) converges in H pp(D E ),

for each data (f u0 u1).

10 Examples

Example 10.1 Let coincide with a connected component of @ D. Then, is a C 1 closed hypersurface in D0 . In this case the analysis above is carried out in the framework of usual (non-weighted) Sobolev spaces on X . For simplicity, we assume that divides D0 into two domains D10 and D20 . Write G1(x y) and G2(x y) for the Green functions of the Dirichlet problem for in D10 and D20 , respectively. Set

G (x y) if (x y) 2 D0 D0 # G (x y) = G12(x y) if (x y) 2 D120 D120 then G (x y), if extended by zero to the whole product (D0 n ) (D0 n ), is easily seen to be the Green function of the Dirichlet problem for in D0 n . On the other hand, the domain D belongs either to D10 or to D20 . Since the

34


potentials Pdl, Psl and G (D ) depend only on the restriction of G (x y) to D D, it suces to make use either of G1(x y) or of G2(x y) to de ne all the potentials. In particular, if = @ D, then we recover the results of Nacinovich and Shlapunov NS96].

Example 10.2 Let P be the gradient operator in Rn. Then ;P P is the

Laplace operator in Rn. Denote by G(x) the standard fundamental solution of convolution type for ;P P , i.e.,

(1=2) log jxj

if n = 2# (1=(2 ; n)n) jxj2;n if n > 2 n being the area of the unit sphere in Rn. If D0 = Rn, n 2, and is the sphere @B (0 R) in Rn, then the re%ection principle yields a Green function of the Dirichlet problem in either of the domains B (0 R) and Rn n B (0 R) in the form G (x y) = G(x ; y) ; (R=jxj)n;2 G ;;(R=jxj)2 x ; y = G(x ; y) ; (R=jyj)n;2 G x ; (R=jyj)2 y

;G(x) =

the symmetry being easily veri ed. Taking B0 = 1 we have n(Pu) = (@=@n)u, the derivative of u along the outward normal vector to the boundary. Thus, if D is a domain in Rn, such that @B (0 R) is a component of the boundary of D, then ;Pdlu (x) = @RD (@=@n) G (x ) u ds Pvf (x) = DR jP=1n (@=@yj ) G(x ) fj dy for x 2 Rn n @ D, where u 2 H 1(D) and f 2 L2(D C n ).


35

References Ant66]

Yu. T. Antokhin, An analytic approach to investigation of equations of the rst type, Dokl. Akad. Nauk SSSR 167 (1966), no. 4, 727{ 730. C22] T. Carleman, Sur la resolution de certaines equations int egrales, Ark. Mat. Astr. Fys. 16 (1922), no. 26, 1{19. DW95] R. Duduchava and W. Wendland, The Wiener-Hopf method for systems of pseudodi erential equations with an application to crack problems, Integral Equ. and Operator Theory 23 (1995), 294{335. Esk73] G. I. Eskin, Boundary Value Problems for Elliptic Pseudodi erential Operators, Nauka, Moscow, 1973. G77] F. D. Gakhov, Boundary Value Problems, Nauka, Moscow, 1977. KGBB76] V. D. Kupradze, T. G. Gegeliya, M. O. Baskheleishvili, and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Nauka, Moscow, 1976. LRS80] M. M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis, Nauka, Moscow, 1980. NS96] M. Nacinovich and A. A. Shlapunov, On iterations of the Green integrals and their applications to elliptic di erential complexes, Math. Nachr. 180 (1996), 243{286. NP91] S. A. Nazarov and B. A. Plamenevskii, Elliptic Boundary Value Problems in Domains with Piecewise Smooth Boundary, Nauka, Moscow, 1991. NCS90] D. Natroshvili, O. Chkadua, and E. Shargorodskii, Mixed problems for homogeneous anisotropic elastic media, Proceedings of I. Vekua Institute of Applied Math., Tbilisi State University 39 (1990), 133{ 181. RST98] V. S. Rabinovich, B.-W. Schulze, and N. N. Tarkhanov, Boundary Value Problems in Cuspidal Wedges, Preprint 24, Univ. Potsdam, Potsdam, October 1998, 66 pp. RS82] S. Rempel and B.-W. Schulze, Index Theory of Elliptic Boundary Problems, Akademie-Verlag, Berlin, 1982.

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Roi96]

Ya. A. Roitberg, Elliptic Boundary Value Problems in Generalized Functions, Kluwer Academic Publishers, Dordrecht NL, 1996. Rom78] A. V. Romanov, Convergence of iterations of Bochner-Martinelli operator, and the Cauchy-Riemann system, Dokl. Akad. Nauk SSSR 242 (1978), no. 4, 780{783. Sch92] B.-W. Schulze, Crack problems in the edge pseudo-di erential calculus, Applic. Analysis 45 (1992), 333{360. Sch98] B.-W. Schulze, Boundary Value Problems and Singular PseudoDifferential Operators, J. Wiley, Chichester, 1998. Sch99] B.-W. Schulze, Pseudo-Di erential Calculus and Applications to Non-Smooth Congurations, Preprint 99/?, Univ. Potsdam, Potsdam, May 1999, 129 pp. Shl99] A. Shlapunov, Iterations of Self-Adjoint Operators and Their Applications to Elliptic Systems, Preprint 99/3, Univ. Potsdam, Potsdam, January 1999, 23 pp. Tar97] N. N. Tarkhanov, The Analysis of Solutions of Elliptic Equations, Kluwer Academic Publishers, Dordrecht, NL, 1997. Tar95] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie-Verlag, Berlin, 1995. Yos65] K. Yosida, Functional Analysis, Springer-Verlag, Berlin et al., 1965. (B.-W. Schulze) Universitat Potsdam, Institut fur Mathematik, Postfach 60 15 53, 14415 Pots-

dam, Germany

E-mail address: [email protected] (A. Shlapunov) Krasnoyarsk State University, pr. Svobodnyi, 79, 660041 Krasnoyarsk, Russia E-mail address: [email protected] (N. Tarkhanov) Institute of Physics, Russian Academy of Sciences, Akademgorodok, 660036

Krasnoyarsk, Russia

E-mail address: [email protected]