JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi Publishing Corporation 486934 10.1155/2013/486934 486934 Research Article The Method of Fischer-Riesz Equations for Elliptic Boundary Value Problems Alsaedy A. Tarkhanov N. Kravchenko Vladislav Institut für Mathematik Universität Potsdam Am Neuen Palais 10 14469 Potsdam Germany uni-potsdam.de 2013 19 2 2013 2013 22 10 2012 30 12 2012 2013 Copyright © 2013 A. Alsaedy and N. Tarkhanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first-order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well-elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.

1. Introduction

The aim of this paper is to bring together two areas in which integral formulas like Green’s formula for harmonic functions are of great importance. The first area is complex analysis where the method of integral representations was a central tool over the second half of the 20th century, see . And the second area is the theory of elliptic boundary problems where the parametrix method led to the most refined results, such as local principle, C*-algebras of pseudodifferential boundary value problems , and so forth.

The method of Fischer-Riesz equations can be specified within a larger approach which is usually referred to as the boundary element method. By this latter is meant a numerical method of solving boundary value problems which have been formulated as boundary integral equations. It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture mechanics, see .

The boundary elements method attempts to use the given boundary conditions and other data of the problem to fit boundary values into the integral equation, rather than values throughout the space defined by a system of partial differential equations. Once this is done, the boundary integral equation can be used again to calculate numerically the solution directly at any desired point in the solution domain. More precisely, from the Cauchy data of the solution on the whole boundary one calculates readily the solution in the domain provided a left fundamental solution of the system is available in an explicit form, see for instance Lemma  10.2.3 in .

The idea of the method of Fischer-Riesz equations goes back at least as far as . The paper  was given by Picone as an invited address before the Second Austrian Mathematical Congress in Insbruck in 1949. He states in the introduction that he asked Fichera to write a certain part of the report. It is a crystallisation in the form of an abstract theory of some of the methods used by the authors and their associates at the National Institute for Applied Mathematics in Rome in the solution of problems involving differential and integro-differential equations. The method is based on some functional interpretation of the relations of mathematical physics analogous to Green’s formula. The central point of the method is a construction of a suitable sequence of functions which are complete in a Lebesgue space L2 on the boundary and satisfy the formal adjoint system in a neighbourhood of the closure of the domain. In  there were no indication to any solution of this problem. In  a general process of constructing a necessary complete sequence of functions was elaborated assuming an explicit fundamental solution of the system. However, this paper fell short of providing a function-theoretic description of the method. In  the second author published a proof of a theorem of functional analysis that had already been obtained at the end of the 1980s. As it became clear later, this theorem was just an abstract exposition of the Fischer-Riesz equations method mentioned in . In  this method was developed for studying the ill-posed Cauchy problem with data on a piece of the boundary for solutions of overdetermined elliptic systems, see also Chapter 11 in .

The purpose of the present paper is to develop the method of Fischer-Riesz equations for general boundary value problems for systems of partial differential equations elliptic in the sense of Douglis-Nirenberg . To escape technicalities related to assigning weights we exploit the result of  and reduce the system to a (possibly overdetermined) first-order system whose classical symbol has a left inverse away from the zero section of the cotangent bundle. In this way we obtain what is often referred to as the first order system with injective symbol. The advantage of such systems lies in the fact that the Cauchy data of a solution just amount to the restriction of the solution to the boundary of the domain. Moreover, to any first order system there corresponds a unique Green operator which leads to a canonical Green formula for solutions. Any normal boundary conditions for solutions of the source system then reduce to an inhomogeneous linear system in the space of Cauchy data.

In contrast to  we elaborate the method of Fischer-Riesz equations for elliptic boundary value problems in Sobolev spaces, for these latter fit well the Fredholm property. If the boundary value problem is Fredholm, then the conditions of solvability obtained by the Fischer-Riesz equations method come to those obtained from the Fredholm theory, that is, the orthogonality to solutions of the homogeneous boundary value problem adjoint relative to the Green formula. The method of Fischer-Riesz equations may then be developed as a tool to get effective approximate solutions, cf. .

2. Reduction to a First-Order System

One of the fundamental problems in the theory of partial differential equations is the problem of classifying equations and systems by type. A specific problem associated with the definition of ellipticity is that when a higher-order equation or system is reduced to a first order system, ellipticity may be destroyed. (We manipulate the concept “elliptic” freely. This concept can be given a strong sense only in an operator algebra with symbol map, where by the ellipticity of an operator is meant the invertibility of its symbol.) One approach to this problem was introduced in the paper  which gave a definition of ellipticity for systems which involved assigning weights to each of the equations and dependent variables and then defining the principal part of the system in terms of those weights. This concept can also be interpreted in terms of generalised homogeneity based on certain group actions in the spaces of preimages and images, see equality (1.4) in . The advantage of the definition of ellipticity given in  is that ellipticity can be preserved, while a higher-order equation or system is reduced to an equivalent first order system. The disadvantage is that the definition is not invariant under nonsingular changes of variables. Therefore, the approach via weights fails to properly recognise elliptic systems. An alternative approach suggested in  is to reduce the original equation or system to an overdetermined first order system and then use the classical symbol, which is natural and invariant way for such systems. In  this result is strengthened by showing that any determined or overdetermined system with smooth coefficients and injective Douglis-Nirenberg symbol can be reduced to an overdetermined first order system with smooth coefficients and injective classical symbol. This reduction is accomplished by introducing as new dependent variables, the derivatives of some of the original variables, and adjoining equations describing the relations between the new variables and the old or among the new variables. Moreover, any overdetermined first order system with smooth coefficients and injective classical symbol can be converted to a determined second-order systems which is elliptic in the sense of Douglis-Nirenberg or under any reasonable definition of ellipticity. The conversion is accomplished by operating on the original system with an appropriately chosen first-order operator. The conversion to a second order system allows the application of the regularity results of . In fact, second order systems are treated in detail in . Note that the systems of partial differential equations usually still contain hidden integrability conditions. The process of their explicit construction is called completion (to involution). In  it is shown that the completion of any determined or overdetermined system with injective Douglis-Nirenberg symbol leads to an equivalent system whose classical symbol has a left inverse. To formulate the main result of  more precisely, we extend the concept of ellipticity in the sense of Douglis-Nirenberg to overdetermined systems.

The systems we consider are of the form (1)A=(Ai,j)i=1,,lj=1,,k, where Ai,j are scalar partial differential operators of order mi,j on an open set 𝒳 in n and lk. (We will generally use the convention that our source system has k dependent variables, l equations, and n independent variables.)

Definition 1.

Suppose there are weights s1,,sl and t1,,tk in , such that mi,jsi+tj. With this structure, the principal symbol of (1) is the matrix (2)σDN(A)(x,ξ)=(σsi+tj(Ai,j)(x,ξ))i=1,,lj=1,,k for (x,ξ)T*𝒳, where σsi+tj(Ai,j) is the homogeneous component of degree si+tj of the full symbol of Ai,j.

System (1) is said to have injective symbol in the sense of Douglis-Nirenberg at x0𝒳 if σDN(A)(x,ξ) has maximal rank (i.e., rank k) for x=x0 and all ξn{0}.

The numbers s1,,sl and t1,,tk are determined uniquely up to an additive constant. Hence, the weights can be normalised by the condition s1,,sl0 and maxsi=0. Then tj satisfy automatically t1,,tk0, for if tj<0, then si+tj<0, and so all the operators A1,j,,Al,j vanish identically. This amounts to saying that the jth dependent variable does not enter into the system, which is impossible.

Let (κλ)λ>0 and (κ~λ)λ>0 be the group actions in k and l, respectively, given by (3)κλ=λ|t|diag(λ-t1,,λ-tk),κ~λ=λ-|s|diag(λs1,,λsl), where |s|=s1++sl and |t|=t1++tk. Then the principal symbol σDN(A) is homogeneous of degree |s|+|t| in the sense that (4)σDN(A)(x,λξ)=λ|s|+|t|κ~λσDN(A)(x,ξ)κλ-1 for all λ>0. For s1==sl=0, one recovers the principal part of A considered by I. G. Petrovskii. For s1==sl=0 and t1==tk=m, one obtains the classical principal symbol. The so-called twisted homogeneity of type (4) is of great importance in the calculus of pseudodifferential operators with operator-valued symbols.

The following result is due to .

Theorem 2.

Any system (1) with coefficients of class Cs,h and injective symbol in the sense of Douglis-Nirenberg in 𝒳 can be converted to an equivalent overdetermined first order system whose coefficients are of class Cs-1,h, and whose classical symbol is injective.

Proof.

As is mentioned in , the reduction procedure used here is related to that attributed to Atiyah and Singer. It is probably not optimal in the sense that it may lead to a first order system which is not the smallest possible representation of the original system.

3. Green Formula

From what has been proved in Section 2 it follows that there is no restriction of generality in assuming that A is a (possibly overdetermined) first order partial differential operator with injective symbol on an open set 𝒳n. Thus, A is of the form (5)A(x,D)=j=1nAj(x)Dj+A0(x), where A1(x),,An(x) and A0(x) are (l×k)-matrices of smooth functions on 𝒳 and Dj=-ıxj with ı=-1. We require (6)σ1(A)(x,ξ):=j=1nAj(x)ξj to have maximal rank (i.e., rank k) for all (x,ξ) away from the zero section of T*𝒳.

In order to get asymptotic results, it is necessary to put some restrictions on A. Our basic assumption is that A satisfies the uniqueness condition of the local Cauchy problem in 𝒳 (condition (U)s, cf. [7, p. 185]). That is, if u is a solution of Au=0 on a connected open set U𝒳 and u vanishes on a nonempty open subset of U, then u0 in U.

Lemma 3.

If A satisfies the condition (U)s in 𝒳, then it has a pseudodifferential left fundamental solution; that is, there is an (k×l)-matrix Φ of classical pseudodifferential operators of order -1 on 𝒳, such that ΦA=I on compactly supported distributions in 𝒳 with values in k.

Proof (see Theorem  4.4.3 of [<xref ref-type="bibr" rid="B23">7</xref>]).

By the very construction, Φ has rational symbol; that is, it satisfies the transmission condition with respect to each hypersurface in 𝒳.

Let 𝒟 be a relatively compact domain with smooth boundary in 𝒳 and B any (k×k)-matrix of smooth functions on the boundary 𝒟 of 𝒟, such that the rank of B(x) is equal to k for all x𝒳. We are interested in the boundary value problem (7)Au=fin  𝒟,Bu=u0at𝒟 with data u0 on 𝒟. The most conventional Hilbert space setting of this problem is H1:=W1,2, hence we choose u0 in H1/2(𝒟,k) and look for a uH1(𝒟,k) satisfying (7).

Lemma 4.

Let C be a ((k-k)×k)-matrix of smooth functions on 𝒟, such that (8)rank(B(x)C(x))=k for all x𝒟. Then there are unique matrices B* and C* of continuous functions on 𝒟 with the property that (9)𝒟((Bu,C*g)x-(Cu,B*g)x)ds=𝒟((Au,g)x-(u,A*g)x)dx for all uH1(𝒟,k) and gH1(𝒟,l), where ds is the surface measure on the boundary.

As usual, we write A* for the formal adjoint of the differential operator A on the open set 𝒳.

Proof.

By assumption, the (k×k)-matrix (10)T(x)=(B(x)C(x)) is invertible for all x𝒟. Write (T(x))-1=(T1(x),T2(x)) where T1 and T2 are (k×k)- and (k×(k-k))-matrices of smooth functions on 𝒟, respectively. The equalities T-1T=Ek and TT-1=Ek amount to T1B+T2C=Ek and (11)BT1=Ek,BT2=0,CT1=0,CT2=Ek-k, where Ek stands for the unity (k×k)-matrix.

Given any uH1(𝒟,k) and gH1(𝒟,l), the Green formula of [7, Section  9.2.2] shows that (12)𝒟(σ(x)u,g)xds=𝒟((Au,g)x-(u,A*g)x)dx, where σ(x) is the principal symbol of A evaluated at the point (x,-ıν(x)) of the complexified cotangent bundle of 𝒳, ν(x) being the outward normal unit vector of the boundary at x𝒟. Substituting u=(T1B+T2C)u into this formula yields (9) with (13)C*=(σT1)*,B*=-(σT2)*, as desired.

From (13) it follows immediately that the rank of C* is equal to k and the rank of B* is k-k.

Elliptic boundary value problems (7) require k=l to be even and k=k/2, in which case also the problem (14)A*g=vin  𝒟,B*g=g0at𝒟 called adjoint to (7) with respect to the Green formula is actually elliptic, cf. .

Given any first order partial differential operators A with injective symbol on 𝒳, the composition Δ=A*A is a second order elliptic operator in the classical sense. This operator is usually referred to as the Laplacian of A. An easy manipulation of Green formula (9) leads to a fairly structural Green formula for the Laplacian Δ.

Theorem 5.

Under the above notation, any functions u,vH2(𝒟,k) satisfy the integral equality (15)𝒟((Bu,C*Av)x-(C*Au,Bv)x-(Cu,B*Av)x+(B*Au,Cv)x)ds=𝒟((Δu,v)x-(u,Δv)x)dx.

Proof.

It suffices to apply (9) twice to the left-hand side of this equality. Cf. Corollary  9.2.12 of .

4. Function Spaces

Denote by H1/2(𝒟,k) the Slobodetskii space of functions of fractional smoothness 1/2 in 𝒟 with values in k, that is, the completion of C(𝒟¯,k) with respect to the norm (16)uH1/2(𝒟,k)=(uL2(𝒟,k)2+𝒟×𝒟|u(x)-u(y)|2|x-y|n+1dxdy)1/2. Obviously, H1/2(𝒟,k) is a Hilbert space. We use the continuous embedding H1/2(𝒟,l)L2(𝒟,l) to specify the dual space of H1/2(𝒟,l) via the pairing in L2(𝒟,l). Namely, let H-1/2(𝒟,l) be the completion of C(𝒟¯,l) with respect to the norm (17)fH-1/2(𝒟,l)=supgC(𝒟¯,l)g0|(f,g)L2(𝒟,l)|gH1/2(𝒟,l).

Using these spaces, we are in a position to enlarge the domain of problem (7). To this end, we write 1/2(𝒟,k) for the completion of C(𝒟¯,k) with respect to the norm (18)u1/2(𝒟,k)=(uH1/2(𝒟,k)2+uL2(𝒟,k)2)1/2.

By the trace theorem, the space 1/2(𝒟,k) contains any space Hs(𝒟,k) with s>1/2. However, the norm of L2(𝒟,k) is not majorised by the norm of H1/2(𝒟,k).

Suppose u1/2(𝒟,k) and {uj} is a sequence in C(𝒟¯,k) converging to u in the norm (18). Then {uj} is a Cauchy sequence in H1/2(𝒟,k), and so it converges to an element uiH1/2(𝒟,k). Moreover, the restrictions of uj to the boundary form a Cauchy sequence in L2(𝒟,k). Hence, the sequence {uj𝒟} converges in the space L2(𝒟,k) to an element ub. It follows immediately that the closure of the mapping u(u𝒟,u𝒟) is an isometry of 1/2(𝒟,k) onto a subspace of the Cartesian product H1/2(𝒟,k)×L2(𝒟,k). For this reason, each element u1/2(𝒟,k) can be identified with its image (ui,ub) in the Cartesian product. We call ubL2(𝒟,k) the (generalised) trace of uiH1/2(𝒟,k) on the boundary of 𝒟 in spite of the fact that the trace ub does not depend continuously on ui.

Lemma 6.

There is a constant c>0, such that (19)AuH-1/2(𝒟,l)cu1/2(𝒟,k) for all uC(𝒟¯,k).

Proof.

The proof is based on manipulations of Green formula (9). See Lemma  2.3.1  in .

It follows from Lemma 6 that the closure A of the mapping uAu for uC(𝒟¯,k) acts continuously from 1/2(𝒟,k) into H-1/2(𝒟,l). Indeed, if u1/2(𝒟,k) and {uj} is a sequence in C(𝒟¯,k) converging to u in 1/2(𝒟,k), then {Auj} is, by Lemma 6, a Cauchy sequence in H-1/2(𝒟,l). Let f be the limit of {Auj} in H-1/2(𝒟,l). It is immaterial which sequence {uj} we choose to define f, and so we may set Au=f. Substituting uj into the estimate of Lemma 6 and letting j, we deduce that this estimate actually holds for all u1/2(𝒟,k). Thus, for each u=(ui,ub) in 1/2(𝒟,k), the element Au is defined in H-1/2(𝒟,l), and the mapping uAu is continuous in the corresponding norms. A passage to the limit similar to the previous implies that Au=f holds for u1/2(𝒟,k) and fH-1/2(𝒟,l) if and only if the couple (ui,ub) satisfy the equation (20)𝒟(ui,A*g)xdx+𝒟((Bub,C*g)x-(Cub,B*g)x)ds=𝒟(f,g)xdx for all gC(𝒟¯,l). In other words, Green’s formula (9) is still valid for functions u1/2(𝒟,k).

If u1/2(𝒟,k) then AuH-1/2(𝒟,l), and so Au can be approximated by functions of Ccomp(𝒟,l) in the H-1/2(𝒟,l)-norm. On multiplying Au by the characteristic function χ𝒟 of 𝒟 we get an element of H-1/2(𝒳,l)   with support in 𝒟¯. It follows that Φ(χ𝒟Au) is well defined and belongs to the local space Hloc1/2(𝒳,k). We now show that any u1/2(𝒟,k) can be restored through the data Au and ub=u𝒟.

To shorten notation we use the same letter Φ(x,y) for the Schwartz kernel of the pseudodifferential operator Φ. Recall that Φ stands for a left fundamental solution of A, see Lemma 3.

Lemma 7.

For each u1/2(𝒟,k), it follows that (21)-𝒟((Bu,C*Φ(x,·)*)y-(Cu,B*Φ(x,·)*)y)ds+𝒟(Au,Φ(x,·)*)ydy={u(x),ifx𝒟,0,if  x𝒳𝒟¯.

Proof.

The proof of Lemma 4 shows that formula (21) is actually equivalent to the equality (22)-Φ([𝒟]σu)+Φ(χ𝒟Au)=χ𝒟u in the sense of distributions on 𝒳, where [𝒟] is the surface layer on 𝒟. This follows in turn from the Green formula and the fact that Φ is a left fundamental solution of A in 𝒳, for (23)Φ([𝒟]σu)=Φ(χ𝒟Au-A(χ𝒟u))=Φ(χ𝒟Au)-χ𝒟u, as desired.

5. Operator-Theoretic Foundations

The operator-theoretic foundations of the method of Fischer-Riesz equations are elaborated in [7, Section  11.1]. It goes back at least as far as . In this section we adapt this method for study of boundary value problem (7) in the Hilbert space 1/2(𝒟,k).

Set (24)H1=1/2(𝒟,k),H=H2L2(𝒟,k-k), where H2=H-1/2(𝒟,l)L2(𝒟,k).

Consider the mapping M:H1H given by Mu=(Au,Bu,Cu), which corresponds to the Cauchy problem for solutions of Au=f in 𝒟 with Cauchy data Bu=u0 and Cu=u1 on 𝒟. By what mentioned previously, M is continuous. In Section 6 we will prove that M has closed range.

Denote by M*:HH1 the operator that is adjoint to M:H1H in the sense of Hilbert spaces.

Lemma 8.

The null-space kerM* of the operator M* is separable in the topology induced from H.

Proof.

This is true by the school fact that any subspace of a separable metric space is separable.

By 𝒮A*(𝒟¯) we denote the space of all infinitely differentiable solutions of the formal adjoint system A*g=0 in a neighbourhood of the closure of 𝒟.

Lemma 9.

Assume that g𝒮A*(𝒟¯). Then the couple (gC*g,B*g) belongs to kerM*.

Proof.

One has to show that (Mu,(gC*g,B*g))H=0 for all uH1. By the Green formula, we get (25)(Mu,(gC*g,B*g))H=𝒟(Au,g)xdx-𝒟((Bu,C*g)x-(Cu,B*g)x)ds=𝒟(u,A*g)xdx=0, as desired.

The subspace of kerM* consisting of all elements of the form (gC*g,B*g), where g𝒮A*(𝒟¯), is separable. Hence, there are many ways to choose a sequence {gi}i=1,2, in 𝒮A*(𝒟¯), such that the system {(giC*gi,B*gi)} is complete in this subspace.

In Example 12 we will show some explicit sequences {gi} with this property. For the moment we fix one of such sequences.

Lemma 10.

As defined above, the system {(giC*gi,B*gi)}i=1,2, is complete in kerM*.

Proof.

Let be a continuous linear functional on kerM* vanishing on each element of the system {(giC*gi,B*gi)}. Since kerM* is a closed subspace of H, the Riesz representation theorem implies the existence of an element (f,u0,u1)kerM*, such that the action of on kerM* consists in scalar multiplication with the element (f,u0,u1). In particular, (26)(giC*gi,B*gi)=𝒟(gi,f)xdx-𝒟((C*gi,u0)x-(B*gi,u1)x)ds=0 for all i=1,2,. Since the system {(giC*gi,B*gi)}i=1,2, is dense in the subspace of kerM* consisting of all elements of the form (gC*g,B*g), where g𝒮A*(𝒟¯), we get (27)𝒟((u0,C*g)x-(u1,B*g)x)ds=𝒟(f,g)xdx for all g𝒮A*(𝒟¯). We now use Theorem 13 which says that there exists a function u1/2(𝒟,k), such that Au=f in 𝒟 and Bu=u0, Cu=u1 at the boundary of 𝒟. In other words, (f,u0,u1)=Mu. Hence it follows that (h)=(h,Mu)H=0 for all hkerM*. Thus, 0, and the standard application of the Hahn-Banach theorem completes the proof.

Write P for the orthogonal projection of H onto its direct summand H2. The composition PM=(A,B) acting from H1 to H2 just amounts to the operator of boundary value problem (7) in the updated setting. More precisely, given any fH-1/2(𝒟,l) and u0L2(𝒟,k), find u1/2(𝒟,k)   satisfying Au=f in 𝒟 and Bu=u0 at 𝒟. The following lemma expresses the most important property of system {gi}.

Lemma 11.

The system {B*gi}i=1,2, is complete in L2(𝒟,k-k) if and only if PM is injective.

Proof.

By the Hahn-Banach theorem, {B*gi} is complete in L2(𝒟,k-k) if and only if any continuous linear functional on L2(𝒟,k-k) vanishing on each element of the system is zero. Pick such a functional . By the Riesz representation theorem there is a function u1L2(𝒟,k-k), such that (h)=(h,u1) for all hL2(𝒟,k-k). So we get (28)((0,0,u1),(giC*gi,B*gi))H=(B*gi,u1)L2(𝒟,k-k)¯=(B*gi)¯=0 for all i=1,2,. Applying Lemma 10 we deduce that the element (0,0,u1) belongs to the orthogonal complement of the subspace kerM* in H. Since the operator M has closed range, the orthogonal complement of kerM* coincides with the range of M. Hence, there is a function u1/2(𝒟,k) satisfying Au=0 in 𝒟 and Bu=0, Cu=u1 at 𝒟. If the operator PM is injective, then u=0 whence u1=0, and =0. Conversely, if the functional is different from zero, then u1 is not zero and so PM fails to be injective, which is precisely the desired conclusion.

After removing the elements which are linear combinations of the previous ones from the system {B*gi}i=1,2,, we get a sequence {gin} in 𝒮A*(𝒟¯), such that the system {B*gin} is linearly independent. Applying then the Gram-Schmidt orthogonalisation to the system {B*gin} in L2(𝒟,k-k), we obtain a new system {en}n=1,2, in 𝒮A*(𝒟¯), such that {B*en} is an orthonormal system in the space L2(𝒟,k-k). Moreover, {B*en} is an orthonormal basis in L2(𝒟,k-k), provided that PM is injective. Note that the elements en of the new system have explicit expressions through the elements {gi1,,gin} of the old system in the form of Gram’s determinants.

Example 12.

Assume that A has real analytic coefficients outside the closure of 𝒟 in 𝒳. Then a familiar trick with the Laplacian A*A shows that A has a left fundamental solution Φ whose Schwartz kernel is real analytic away from the diagonal of (𝒳𝒟¯)×(𝒳𝒟¯). Let {xi} be a finite or countable set of points in 𝒳𝒟¯, such that each connected component of 𝒳𝒟¯ contains at least one point xi. Then the columns of DxαΦ(xi,·)* belong to 𝒮A*(𝒟¯), and the system {B*DxαΦ(xi,·)*} is complete in the subspace of L2(𝒟,k-k) formed by elements of the type {B*g} with g𝒮A*(𝒟¯).

The proof of this fact actually repeats the reasoning of Example  11.4.14 in . Apparently the system of Example 12 is most convenient for numerical simulations.

6. The Cauchy Problem

The Green formula (9) displays the Cauchy data of uH1(𝒟,k) at the boundary of 𝒟 with respect to the operator A. These are Bu and Cu at 𝒟. Hence we formulate the Cauchy problem as follows: given any fH-1/2(𝒟,l)u0L2(𝒟,k), and u1L2(𝒟,k-k), find a function u1/2(𝒟,k), satisfying Au=f in 𝒟 and (29)Bu=u0,Cu=u1 at 𝒟.

The Cauchy problem for solutions of systems with injective symbol and data on the whole boundary was intensively studied in the 1960s. This study was motivated to a certain extent by . For a recent account of the theory we refer to [21, 22].

Theorem 13.

Let fH-1/2(𝒟,l),u0L2(𝒟,k), and u1L2(𝒟,k-k) be given functions. In order that there might exist a solution u1/2(𝒟,k) to the system Au=f in 𝒟 that is subject to boundary conditions (29), it is necessary and sufficient that (30)𝒟((u0,C*g)x-(u1,B*g)x)ds=𝒟(f,g)xdx for all g𝒮A*(𝒟¯).

Proof. Necessity.

If u1/2(𝒟,k) is a solution of the Cauchy problem with data f,u0, and u1, then by the Green formula (31)𝒟((u0,C*g)x-(u1,B*g)x)ds=𝒟((Bu,C*g)x-(Cu,B*g)x)ds=𝒟((Au,g)x-(u,A*g)x)dx=𝒟(f,g)xdx for all g𝒮A*(𝒟¯), as required.

Sufficiency. We introduce a function U in 𝒳𝒟 with values in k by the Green-type integral as follows: (32)U(x)=-𝒟((u0,C*Φ(x,·)*)y-(u1,B*Φ(x,·)*)y)ds+𝒟(f,Φ(x,·)*)ydy, where x𝒳𝒟. An easy calculation using (13) shows that (33)(u0,C*Φ(x,·)*)y-(u1,B*Φ(x,·)*)y=Φ(x,·)σub on 𝒟, where (34)ub=T-1(u0u1). It is clear that ub is of class L2(𝒟,k) if and only if u0L2(𝒟,k) and u1L2(𝒟,k-k). Thus, formula (32) reduces to (35)U=-Φ([𝒟]σub)+Φ(χ𝒟f) in 𝒳𝒟.

For each fixed x𝒳𝒟¯, the columns of the matrix Φ(x,·)* belong to 𝒮A*(𝒟¯). Hence, (30) implies that U vanishes in the complement of 𝒟¯.

Set u=U𝒟. We next prove that u is the desired solution of the Cauchy problem. This is equivalent to saying that u1/2(𝒟,k),  Au=f in 𝒟, and u𝒟=ub at 𝒟.

By Lemma  3.2 of , the double layer potential Φ([𝒟]σub) in 𝒟 belongs to 1/2(𝒟,k). Moreover, the volume potential Φ(χ𝒟f) in 𝒟 is of class 1/2(𝒟,k), which is due to Lemma 6 ibid. Hence it follows that u1/2(𝒟,k). Be the function f zero, we would be able to deduce the rest of the proof from Theorem  10.3.4 of .

In the general case we complete Φ to a fundamental solution at step 0 of a compatibility complex of A, cf. Theorem  4.4.3 of . An easy computation using solvability condition (30) yields (36)AU=-[𝒟]σub+χ𝒟f in the sense of distributions in 𝒳. In particular, Au=f in 𝒟.

Since ubL2(𝒟,k), the jump of the double layer potential Φ([𝒟]σub) under crossing the surface 𝒟 from 𝒳𝒟¯ to 𝒟 just amounts to ub. This is true even for all distributions ub on 𝒟 taking their values in k, see Theorem  10.1.5 in . For the square integrable densities ub the jump is understood in an appropriate sense including the L2(𝒟,k)-norm.

On the other hand, the volume potential Φ(χ𝒟f) has no jump at the boundary of 𝒟, for χ𝒟fH𝒟¯-1/2(𝒳,k). Summarising, we conclude that u𝒟=ub, for U vanishes in 𝒳𝒟¯. For a thorough treatment of this equality we refer the reader to Theorem  4.3 of .

7. The Fischer-Riesz Equations

Let {gi}i=1,2, be an arbitrary sequence in 𝒮A*(𝒟¯) with the property that the system {(giC*gi,B*gi)} is complete in kerM*. Applying the Gram-Schmidt orthogonalisation to the system {B*gi} in L2(𝒟,k-k), we obtain a new system {en}n=1,2, in 𝒮A*(𝒟¯), such that the system {B*en} is orthonormal in the space L2(𝒟,k-k).

Given any u1L2(𝒟,k-k), we denote by kn(u1) the Fourier coefficients of u1 with respect to the system {B*en}; that is, (37)kn(u1)=𝒟(u1,B*en)yds for n=1,2,.

Lemma 14.

If u1/2(𝒟,k), then (38)kn(Cu)=𝒟(Bu,C*en)yds-𝒟(Au,en)ydy, where n=1,2,.

Proof.

Using Lemma 9, we obtain (39)kn(Cu)=𝒟(Cu,B*en)yds-(Mu,(enC*en,B*en))H=𝒟(Bu,C*en)yds-𝒟(Au,en)ydy, as desired.

Thus, in order to find the Fourier coefficients of the data Cu on the boundary with respect to the system {B*en} in L2(𝒟,k-k), it suffices to know only the data Au and Bu of problem (7).

Theorem 15.

Let fH-1/2(𝒟,l) and u0L2(𝒟,k). In order that there is a u1/2(𝒟,k), such that Au=f in 𝒟 and Bu=u0 at 𝒟, and it is necessary and sufficient that

n=1|cn|2<, where cn=𝒟(u0,C*en)yds-𝒟(f,en)ydy,

𝒟(u0,C*g)yds-𝒟(f,g)ydy=0 for all g𝒮A*(𝒟¯) satisfying B*g=0 at the boundary.

Proof. Necessity.

Suppose there is a function u1/2(𝒟,k) satisfying Au=f in 𝒟 and Bu=u0 at 𝒟. Then cn=kn(Cu) for all n=1,2,, which is due to Lemma 14. Applying the Bessel inequality yields (40)n=1|cn|2=n=1|kn(Cu)|2𝒟|Cu|2ds<, and (1) is proved. On the other hand, (2) follows immediately from the Green formula, as desired.

Sufficiency. We now assume that (1) and (2) are satisfied. Condition (1) implies, by the Fischer-Riesz theorem, that there exists a function u1L2(𝒟,k-k), such that {cn}n=1,2, are the Fourier coefficients of u1 with respect to the orthonormal system {B*en} in L2(𝒟,k-k). In other words, we get cn=kn(u1) for all n=1,2,. On substituting formulas for cn from (1) to these equalities we arrive at the orthogonality relations (41)𝒟(f,en)ydy-𝒟((u0,C*en)y-(u1,B*en)y)ds=0 for n=1,2,, cf. (30).

Our next goal is to prove that the element (f,u0,u1)H is actually orthogonal to all elements of the system {(giC*gi,B*gi)}i=1,2, in H, this latter being complete in kerM*. To do this, let us recall how the system {en} has been obtained from the system {gi}.

Even if the system {(giC*gi,B*gi)} is linearly independent in H, the system {B*gi} may have elements which are linear combinations of the previous ones in the space L2(𝒟,k-k). Such elements should be eliminated from the system before applying the Gram-Schmidt orthogonalisation.

For example, suppose that, for some i, the equality (42)B*gi=j=1i-1ci,jB*gj is fulfilled with suitable complex numbers ci,j. Consider the function (43)gi=gi-j=1i-1ci,jgj which belongs to 𝒮A*(𝒟¯). Obviously, (giC*gi,B*gi) lies in kerM* and satisfies B*gi  =0. It follows that (44)gi=j=1i-1ci,jgj+gi. All the other elements (giC*gi,B*gi), except for the eliminated ones, are expressed, by the contents of Gram-Schmidt orthogonalisation, as linear combinations of the elements {(enC*en,B*en)}n=1,,i. Thus, any element of the system {(giC*gi,B*gi)} has a unique expression through the elements of the system {(enC*en,B*en)}n=1,2, in the form (45)gi=n=1ici,nen+gi, where gi𝒮A*(𝒟¯) satisfies B*gi=0 at the boundary 𝒟.

From equalities (41) and (45) and condition (2) of the theorem it follows immediately that (46)((f,u0,u1),(giC*gi,B*gi))H=n=1ici,n((f,u0,u1),(enC*en,B*en))H+((f,u0,u1),(giC*gi,B*gi))H=0 for all i=1,2,. Since the system {(giC*gi,B*gi)}i=1,2, is complete in kerM*, the element (f,u0,u1) belongs to the orthogonal complement of this subspace in H. Using the lemma of operator kernel annihilator, we deduce that there exists a function u1/2(𝒟,k) satisfying Mu=(f,u0,u1). In particular, Au=f in 𝒟 and Bu=u0 at 𝒟; that is, u is the desired solution of boundary value problem (7).

The convergence of the series in (1) guarantees the stability of boundary value problem (7). Under this condition, the range of the mapping PM is described in terms of continuous linear functionals on the space H, cf. (2), which is impossible in the general case.

Corollary 16.

Under the hypotheses of Theorem 15, if moreover the homogeneous adjoint boundary value problem (14) has no smooth solutions in 𝒟¯ different from zero, then for problem (7) to have a solution u1/2(𝒟,k) it is necessary and sufficient that (47)n=1|cn|2<.

Proof.

This follows immediately from Theorem 15 since condition (2) is automatically fulfilled.

8. Regularisation of Solutions

Note that the proof of Theorem 15 works without the assumption that the operator PM in H is injective. Our next objective will be to construct an approximate solution to boundary value problem (7). To this end it is natural to assume that the homogeneous boundary value problem corresponding to (7) has only zero solution in the space 1/2(𝒟,k); that is, the mapping PM is injective. In this case the orthonormal system {B*en} is actually complete in the space L2(𝒟,k-k). The orthonormal bases in L2(𝒟,k-k) of this form are said to be special, cf. , [7, Section  11.3].

For x𝒳𝒟, we denote by kn(B*Φ(x,·)*)   the k-row whose entries are the Fourier coefficients of the columns of the ((k-k)×k)-matrix B*Φ(x,·)* with respect to the orthonormal basis {B*en}n=1,2, in L2(𝒟,k-k). More precisely, we set (48)kn(B*Φ(x,·)*)=𝒟(B*Φ(x,·)*,B*en)yds for n=1,2,.

Lemma 17.

For n=1,2,, the coefficients kn(B*Φ(x,·)*) are infinitely differentiable functions in 𝒳𝒟 with values in (k)*.

Proof.

The assertion is obvious, for the fundamental solution Φ(x,y) is C away from the diagonal of 𝒳×𝒳.

Consider the following (Schwartz) kernels RN defined for x𝒳𝒟 and y in a neighbourhood of 𝒟¯: (49)RN(x,y)=Φ(x,y)-n=1Nkn(B*Φ(x,·)*)*en(y)*, where N=1,2,.

Lemma 18.

As defined above, the kernels RN are C in x𝒳𝒟 and y in a neighbourhood of 𝒟¯ except for the diagonal {x=y}, and A*(y,D)RN(·,y)*=0 on this set.

Proof.

This follows immediately from Lemma 17 and the fact that en𝒮A*(𝒟¯), as desired.

The sequence {RN} provides a very special approximation of the fundamental solution Φ.

Lemma 19.

The sequence {B*RN(x,·)*}N=1,2, converges to zero in the norm of L2(𝒟,(k-k)×k) uniformly in x on compact subsets of 𝒳𝒟.

Proof.

In fact, we get (50)B*RN(x,·)*=B*Φ(x,·)*-n=1NB*enkn(B*Φ(x,·)*)=n=N+1B*enkn(B*Φ(x,·)*) for each fixed x𝒳𝒟. The right-hand side of this equality is a remainder of the Fourier series of the element B*RN(x,·)* with respect to the orthonormal basis {B*en} in L2(𝒟,k-k). Hence, it tends to zero in the L2(𝒟,(k-k)×k)-norm, as N. This proves the first part of the lemma. The second part follows from a general remark on Fourier series, that the mapping of 𝒳𝒟 to L2(𝒟,(k-k)×k) given by xB*Φ(x,·)* is continuous.

The convergence of the approximations allows one to reconstruct solutions u of the class 1/2(𝒟,k) through their data Au and Bu.

Theorem 20.

Every function u1/2(𝒟,k) can be represented by the integral formula (51)u(x)=limN(-𝒟(Bu,C*RN(x,·)*)yds+𝒟(Au,RN(x,·)*)ydy) for all x𝒟.

Proof.

Fix a point x𝒟. Since RN(x,·)* and Φ(x,·)* differ by a k-row of smooth solutions of the system A*g=0 in a neighbourhood of 𝒟¯, one can write by the Green formula (52)u(x)=-𝒟((Bu,C*RN(x,·)*)y-(Cu,B*RN(x,·)*)y)ds+𝒟(Au,RN(x,·)*)ydy for any N=1,2,. From u1/2(𝒟,k) we deduce that CuL2(𝒟,k-k). Hence it follows by Lemma 19 that (53)limN𝒟(Cu,B*RN(x,·)*)yds=0. Thus, letting N in (52) establishes the formula.

As mentioned, several problems of mathematical physics formulas for approximate solution like that of Theorem 20 were earlier obtained by Kupradze, see . In the Cauchy problem for solutions of overdetermined elliptic systems with data on an open part of the boundary an analogous formula was derived in . Chapter 11 in  contains many explicit applications of the formula of . To the best of our knowledge, Theorem 20 is new for general boundary value problems (7).

9. Solvability of Elliptic Boundary Value Problems

We can now return to the classical setting of boundary value problem (7) which is H1=H1(𝒟,k). Given any uH1(𝒟,k), both Au and Bu are well defined in L2(𝒟,l) and H1/2(𝒟,k), respectively. Hence, the analysis does not require any function spaces of negative smoothness but distributions. More generally, let s be a natural number. Given any fHs-1(𝒟,l) and u0 in Hs-1/2(𝒟,k), we look for a uHs(𝒟,k) satisfying (7). Theorem 15 still applies to establish the existence of a weak solution u1/2(𝒟,k), if the conditions (1) and (2) are fulfilled. To infer the existence of a classical solution, one needs a regularity theorem for weak solutions in 1/2(𝒟,k) saying that any weak solution is actually a classical one; that is, uHs(𝒟,k) provided AuHs-1(𝒟,l) and BuHs-1/2(𝒟,k). This is the case if (7) is an elliptic boundary value problem; that is, A is elliptic (l=k), and the pair (A,B) satisfies the Shapiro-Lopatinskii condition at the boundary of 𝒟, see Section  10.5 in . For general operators A with injective symbol the regularity problem may be reduced to a regularity theorem for weak solutions of A*Au=A*f in 𝒟 with boundary data Bu=u0 and B*Au=B*f at 𝒟, see Lemma 4.

Corollary 21.

Suppose a regularity theorem holds for (7). Let fHs-1(𝒟,l) and u0Hs-1/2(𝒟,k)  , where s=1,2,. Then, in order that there is a uHs(𝒟,k), such that Au=f in 𝒟 and Bu=u0 at 𝒟, and it is necessary and sufficient that

n=1|cn|2<, where cn=𝒟(u0,C*en)yds-𝒟(f,en)ydy,

𝒟(u0,C*g)yds-𝒟(f,g)ydy=0 for all g𝒮A*(𝒟¯) satisfying B*g=0 at the boundary.

Proof.

It is sufficient to prove the sufficiency of conditions (1) and (2). If the conditions (1) and (2) are satisfied, then there exists a function u1/2(𝒟,k), such that Au=f in 𝒟 and Bu=u0 at 𝒟. Since AuHs-1(𝒟,l) and BuHs-1/2(𝒟,k), the regularity theorem implies that uHs(𝒟,k), as desired.

If (7) is elliptic, then the problem (A*,B*) is adjoint to (7) with respect to the Green formula. By the Fredholm property, the space of all g𝒮A*(𝒟¯) satisfying B*g=0 at 𝒟 is finite dimensional. Moreover, the condition (2) alone is sufficient for the existence of a solution uHs(𝒟,k) to problem (7). Hence it follows that for elliptic boundary value problems the condition (1) is automatically fulfilled. In other words, if boundary value problem (7) is elliptic, then condition (2) is necessary and sufficient for a solution to exist. If moreover each g𝒮A*(𝒟¯) satisfying B*g=0 at the boundary vanishes, then problem (7) is solvable for all data fL2(𝒟,l) and u0H1/2(𝒟,k). Condition (1) is of purely theoretical interest, for there is no efficient way to verify it.

Thus, the regularity problem for weak solutions of (7) is still of primary character in the study of boundary value problems. On the other hand, our approach demonstrates rather strikingly that Theorem 20 is of great importance for numerical simulation.

Corollary 21 applies in particular to boundary value problems for generalised Cauchy-Riemann systems in the space , see also [18, 26].

Acknowledgment

The research of the first author was supported by the German Academic Exchange Service (DAAD).

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