IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 10.1155/2014/245350 245350 Research Article Mixed Boundary Value Problem on Hypersurfaces DuDuchava R. Tsaava M. Tsutsunava T. Stratis Ioannis G. 1 A. Razmadze Mathematical Institute Tbilisi State University Tamarashvili Street 6, 0177 Tbilisi Georgia tsu.edu.ge 2014 1782014 2014 22 02 2014 31 05 2014 02 06 2014 17 8 2014 2014 Copyright © 2014 R. DuDuchava et al. 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.

The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation divC(ACφ)=f on a smooth hypersurface C with the boundary Γ=C in Rn. A(x) is an n×n bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts Γ=ΓDΓN and on ΓD the Dirichlet boundary conditions are prescribed, while on ΓN the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to divS(AS) is proved, which is interpreted as the invertibility of this operator in the setting Hp,#s(S)Hp,#s-2(S), where Hp,#s(S) is a subspace of the Bessel potential space and consists of functions with mean value zero.

1. Introduction

Let CS be a smooth subsurface of a closed hypersurface S in the Euclidean space Rn (see Section 2 for details) and let Γ=C be its smooth boundary C=Γ. Let Dj:=j-νjν,j=1,,n, be Günter’s tangential derivatives, and let ΔC(t,D):=D12++Dn2 be the Laplace-Beltrami operator restricted to the hypersurface C (see  and Section 2 below for details).

In , the boundary value problem for the Laplace-Beltrami equation with the Dirichlet boundary condition (1)(ΔC(t,D)u)(t)=f(t),tC,u+(s)=g(s),onΓ and with the Neumann boundary condition (2)(ΔC(t,D)u)(t)=f(t),tC,(DνΓu)+(s)=h(s),onΓ were considered where νΓ:=(νΓ,1,,νΓ,n) is the unit normal vector field to the boundary Γ and tangent to the hypersurface C and φ+ denotes the trace on the boundary. The derivative (3)DνΓ:=k=1nνΓ,kDk,sΓ, is tangent to the hypersurface C and normal with respect to the boundary Γ.

The BVPs (1) and (2) were investigated in  in the following classical weak setting: (4)fH~-1(C),gH1/2(Γ),hH-1/2(Γ), and also in nonclassical weak setting: (5)fH~ps-2(C),gWps-1/p(Γ),hWps-1-1/p(Γ),nnnnnnnnnnnnnnnnnnnnnnnnnn1<p<,s>1p, and the following was proved.

Theorem 1.

The Dirichlet problems (1), (4) and (1), (5) have a unique solution.

For the solvability of the Neumann problems (2), (4) and (2), (5), the necessary and sufficient compatibility condition (6)(f,1)C-(h,1)Γ=0 should be fulfilled, which guarantees the existence and the uniqueness of solution.

If f and h are regular integrable functions, the compatibility condition (6) acquires the form (7)Cf(y)dσ-Γh(s)ds=0.

In Remarks 15 and 16, it is shown that the unique solvability of the Dirichlet BVP (1), (4) and the Neumenn BVP (2), (4) in the classical formulation follows from the Lax-Milgram Lemma.

The investigation in  is based on the technique of Günter’s derivatives developed in the preprint of Duduchava from 2002 and later in the paper of Duduchava et al.  and applies potential method. Similar problems, for p=2, by different technique were investigated earlier in the paper of Mitrea and Taylor .

The purpose of the present paper is to investigate the boundary value problems for the anisotropic Laplace equation with mixed boundary conditions: (8)divC(ACu)(t)=f(t),tC,u+(s)=g(s),onΓD,νΓ(s),(ACu)+(s)=h(s),onΓN, where C=Γ=ΓDΓN is a decomposition of the boundary into two connected parts, A={aij} is an n×n strictly positive definite matrix, and (9)A(x)ξ,ξCξ2,ξRn, for all xC. We consider the BVP (8) in the weak classical setting (4).

The nonclassical weak setting (5) will be considered in a forthcoming paper.

Remark 2.

As shown in , page 196, condition (4) does not ensure the uniqueness of solutions to the BVPs (1), (2) and (8). The right hand side f needs additional constraint that it belongs to the subspace H~0-1(Ω)H~-1(Ω) which is the orthogonal complement to the subspace H~-1(Γ) of those distributions from H~-1(Ω) which are supported on the boundary Γ=Ω of the domain only.

For the non-classical setting (5) we schould impose a similar constraint fH~p,0-1(Ω)H~p-1(Ω), which is defined as in the case p=2.

For the classical setting (4), we apply the Lax-Milgram Lemma and prove unique solvability of the problem rather easily, while (5) in the nonclassical investigation relies again on the potential method.

Mixed BVPs for the Laplace equation in domains were investigated by Lax-Milgram Lemma by many authors (see, e.g., the recent lecture notes online ).

BVPs on hypersurfaces arise in a variety of situations and have many practical applications. See, for example, [6, Section 7.2] for the heat conduction by surfaces, [7, Section 10] for the equations of surface flow, [8, 9] for the vacuum Einstein equations describing gravitational fields, and  for the Navier-Stokes equations on spherical domains, as well as the references therein.

A hypersurface S in Rn has the natural structure of an (n-1)-dimensional Riemannian manifold and the aforementioned PDEs are not the immediate analogues of the ones corresponding to the flat, Euclidean case, since they have to take into consideration geometric characteristics of S such as curvature. Inherently, these PDEs are originally written in local coordinates, intrinsic to the manifold structure of S.

Another problem considered in the present paper is the existence of a fundamental solution for the Laplace-Beltrami operator. An essential difference between differential operators on hypersurfaces and the Euclidean space Rn lies in the existence of fundamental solution: in Rn fundamental solution exists for all partial differential operators with constant coefficients if it is not trivially zero. On a hypersurface even Laplace-Beltrami operator does not have a fundamental solution because it has a nontrivial kernel, constants, in all Bessel potential spaces. Therefore we consider Laplace-Beltrami operator in Hilbert spaces with detached constants Wp,#s(S)Wp,#s-2(S), for all 1<p<,sR, and prove that it is an invertible operator. Another description of the space Wp,#s(S) is that it consists of all functions φWps(S) (distributions if s<0, which have the zero mean value, (φ,1)S=0). The established invertibility implies the existence of the certain fundamental solution, which can be used to define the volume (Newtonian), single layer, and double layer potentials.

The structure of the paper is as follows. In Section 2, we expose all necessary definitions and some auxiliary material, partly new ones. Here the invertibility of the Laplace-Beltrami operator in the setting Wp,#s(S)Wp,#s-2(S) is proved. In Section 3, using the Lax-Milgram Lemma, it is proved that the basic mixed BVP (8) has a unique solution in the weak classical setting (4).

2. Auxiliary Material

We commence with definitions of a hypersurface. There exist other equivalent definitions but these are the most convenient for us. Equivalence of these definitions and some other properties of hypersurfaces are exposed, for example, in [3, 11].

Definition 3.

A subset SRn of the Euclidean space is called a hypersurface if it has a covering S=j=1MSj and coordinate mappings (10)Θj:ωjSj:=Θj(ωj)Rn,nnnlnlωjRn-1,j=1,,M, such that the corresponding differentials (11)DΘj(p):=matr[1Θj(p),,n-1Θj(p)] have the full rank (12)mmmmmmmmrankDΘj(p)=n-1,pYj,k=1,,n,j=1,,M; that is, all points of ωj are regular for Θj for all j=1,,M.

Such a mapping is called an immersion as well.

Here and in what follows matr[x1,,xk] refers to the matrix with the listed vectors x1,,xk as columns.

A hypersurface is called smooth if the corresponding coordinate diffeomorphisms Θj in (10) are smooth (C-smooth). Similarly is defined a μ-smooth hypersurface.

The next definition of a hypersurface is implicit.

Definition 4.

Let k1 and ωRn be a compact domain. An implicit Ck-smooth hypersurface in Rn is defined as the set (13)S={Xω:ΨS(X)=0}, where ΨS:ωR is a Ck-mapping, which is regular Ψ(X)0.

Stoke’s derivatives are concrete examples of tangential operators (14)MS:=[Mjk]n×n,Mjk:=νjk-νkj=rtialmj,k.

Corollary 5 (Green’s formula; cf. [<xref ref-type="bibr" rid="B5">1</xref>]).

Let ΩRn be a domain with Lipschitz boundary. For the Laplace operator(15)Δ:=12++n2 and functions φ,ψW21(Ω),ΔψW~1(Ω), the following I and II Green formulae are valid: (16)Ω(Δψ)(y)φ(y)dy=Ω(νψ)(τ)φ(τ)dσ-j=1nΩ(jψ)(y)(jφ)(y)dy,Ω(Δψ)(y)φ(y)dy=Ωψ(y)(Δφ)(y)dy+Ω[(νψ)(τ)φ(τ)+ψ(τ)(νφ)(τ)]dσ. Integrals in (16) are understood in the sense of duality between spaces W~-1(Ω) and W1(Ω), H-1/2(Ω) and H1/2(Ω), and so forth.

Let S be a closed hypersurface in Rn and let C be a smooth subsurface of S, given by an immersion (17)Θ:ωC,ωRn-1 with a boundary Γ=C, given by another immersion (18)ΘΓ:ωΓ:=C,ωRn-2, and let ν(X) be the outer unit normal vector field to C and let N(x) denote an extended unit field in a neighborhood ωC of C. νΓ(t) is the outer normal vector field to the boundary Γ, which is tangential to C.

A curve on a smooth surface C is a mapping (19)γ:IC,I:=(a,b]R, of a line interval I to C.

A vector field on a domain Ω in Rn is a mapping (20)U:ΩRn,U(x)=j=1nUj(x)ej, where UjC0(Ω) and ej is the element of the natural Cartesian basis in Rn: (21)e1:=(1,0,,0),,en:=(0,,0,1), in the Euclidean space Rn.

By V(Ω) we denote the set of all smooth vector fields on Ω.

A vector field UV(Ω) defines the first order differential operator(22)Uf(x)=rtialUf(x):=limh0f(FUh(x))-f(x)h=ddtf(FUt(x))|t=0, where FUt(x) is the orbit of the vector field U.

Let (23)P(D)u=j=1najju+bu,aj,bC1(Rm×m), be a first order differential operator with real valued (variable) matrix coefficients, acting on vector-valued functions in Rn, and its principal symbol is given by the matrix-valued function (24)σ(P;ξ):=j=1najξj,ξ={ξj}j=1nRn.

To distinguish an open and a closed hypersurface, we use the notation S for a closed hypersurface without the boundary rtialS= (we remind the reader that the notation C is reserved for an open hypersurface with the boundary Γ:=rtialS).

Definition 6.

We say that P is a tangential operator to the hypersurface S, with unit normal ν, if (25)σ(P;ν)=0on  the  hypersurfaceS.

Lemma 7.

Let P be, as in (23), a first order differential operator with C1-smooth coefficients. P is tangential if and only if the adjoint P* operator is tangential.

If P is tangential to S and P is defined in a neighborhood of S, then (26)(Pφ)S=P(φS), for every C1 function φ defined in a neighborhood of S.

We continue with the definition of the surface divergence divS, the surface gradient S, and the surface Laplace-Beltrami operator ΔS.

According to the classical differential geometry, the surface gradient S of a function fC1(S) is defined by (27)Sf=gradf=j,k(gjkrtialjf)rtialrtialxk, and the surface divergence of a smooth tangential vector field V is defined by (28)divSV:=k=1n-1V;jj,V;kj:=kVj+m=1n-1ΓkmjVm, where Γkmj denotes the Christoffel symbols(29)Γkmj:=12k=1n-1gjl[mgkl+kgml-lgkm]=Γmkj and G:=[gjk] is the covariant Riemann metric tensor, while G-1:=[gjk] is the inverse to it-the contravariant Riemann tensor.

div S is the negative dual to the surface gradient: (30)divSV,f:=-V,Sf,VV(S),fC1(S). The Laplace-Beltrami operator ΔS on S is defined as the composition (31)ΔSψ=divSSψ=-S*(Sψ).

Theorem 8 (cf. [<xref ref-type="bibr" rid="B8">2</xref>]).

For any function φC1(S), one has (32)Sφ={D1φ,D2φ,,Dnφ}. Also, for a 1-smooth tangential vector field V=j=1nVjejV(S), (33)divSV=-S*V:=j=1nDjVj. The Laplace-Beltrami operator ΔS on S takes the form (34)ΔSψ=j=1nDj2ψ=j<kMjk2ψ=12j,k=1nMjk2ψ,ψC2(S).

Corollary 9 (cf. [<xref ref-type="bibr" rid="B8">2</xref>]).

Let S be a smooth closed hypersurface. The homogeneous equation (35)ΔSψ=0 has only a constant solution in the space W1(S).

Proof.

Due to (31) and (35), we get (36)0=(-ΔSψ,ψ)=(Sψ,Sψ)=SψL2(S), which gives Sψ=0. But the trivial surface gradient means constant function ψ=const (this is easy to ascertain by analysing the definition of Günter’s derivatives; see, e.g., ).

Let M be a nontrivial mesM, smooth closed or open hypersurface, sR, and 1<p<. For the definitions of Bessel potential Hps(M) and Sobolev-Slobodeckii Wps(M) spaces for a closed smooth manifold M, we refer to  (also see [1, 13, 14]). For p=2 the Sobolev-Slobodetski W2s(M) and Bessel potential H2s(Γ) spaces coincide (i.e., the norms are equivalent).

Let C be a subsurface of a smooth closed surface M, CM, with the smooth boundary Γ:=rtialS. The space H~ps(C) is defined as the subspace of those functions φHps(M), which are supported in the closure of the subsurface, suppφC¯, whereas Hps(C) denotes the quotient space Hps(C)=Hps(M)/H~ps(Cc), and Cc:=MC¯ is the complementary subsurface to C. The space Hps(C) can be identified with the space of distributions φ on C which have an extension to a distribution lφHps(M). Therefore, rCHps(M)=Hps(C), where rC denotes the restriction operator of functions (distributions) from the surface M to the subsurface Rn.

The spaces W~ps(C) and Wps(C) are defined similarly (see  and also [1, 13, 14]).

By Xps(M) we denote one of the spaces: Hps(M) and Sobolev-Slobodecki Wps(M) (if M is closed or open), and by X~ps(M) denote one of the spaces: H~ps(M) and W~ps(M) (if M is open). Consider the space (37)Xp,#s(M):={φX2s(M):(φ,1)=0}. It is obvious that Xp,#s(M) does not contain constants: if c0=constXp,#s(M), then (38)0=(c0,1)=c0(1,1)=c0mesM and c0=0. Moreover, Xps(M) decomposes into the direct sum (39)Xps(M)=Xp,#s(M)+{const} and the dual (adjoint) space is (40)(Xp,#s(M))*=Xp,#-s(M),p:=pp-1. In fact, the decomposition (39) follows from the representation (41)φ=φ0+φaver,φ0Xp,#s(M),φaver:=1mesM(φ,1) of arbitrary function φXps(M), because the average of the difference of a function and its average is zero: (φ0)aver=(φ-φaver)aver=0.

Since the Sobolev space Wp,#m(M) with integer smoothness parameter m=1,2, does not contain constants, due to Corollary 9 the equivalent norm in this space can also be defined as follows: (42)φWp,#m(M)0:=1|α|mDαφLp(M). In particular, in the space Wp,#1(M) the equivalent norm is (43)φWp,#1(M)0:=SφLp(M).

The description (40) of the dual space follows from the fact that the dual space to Xps(M) is Xp-s(M) (see ) and, therefore, due to the decomposition (39) and Hahn-Banach theorem the dual space to Xp,#s(M) should be embedded into Xp-s(M). The only functional from Xp-s(M) that vanishes on the entire space Xp,#s(M) is constant 1Xp-s(M) (see definition (37)). After detaching this functional the remainder coincides, due to (39), with the space Xp,#-s(M), which is the dual to Xp,#s(M).

Theorem 10.

Let S be l-smooth l=1,2,,1<p< and |s|l. Let Xps(S) be the same as in (37)–(40).

Let HCl-1(Rn) have one of the following properties:

H has a nonnegative real part ReH(t)0, for all tS, and messuppReH0;

H has a constant complex part ImH(t)=const0;

ReH=0, messuppImH0, and the complex part ImH does not change the sign: either ImH(t)0, for all tS, or ImH(t)0, for all tS.

The perturbed operator (44)divSAS-HI:Xps+1(S)Xps-1(S) is invertible, which can be interpreted as the existence of the fundamental solution to divSAS-HI.

The operator divSAS itself is invertible between the spaces with detached constants (see (37)) (45)divSAS:Xp,#s+1(S)Xp,#s-1(S). And, therefore, divSAS has the fundamental solution in the setting (45).

Proof.

The first part of the theorem is proved in [3, Theorem 7.1] for the space setting W1(S)W-1(S) only. Therefore, we will prove it here in full generality.

First of all, note that the operator (44) is bounded and elliptic, as an elliptic operator on the closed hypersurface divSAS-HI in (44) is Fredholm, for all sR and 1<p< (it has a parametrix if S is infinitely smooth; see [13, 15, 16]). On the other hand, (46)(-(divSAS-H)φ,φ)L2(S)=-Γ(νΓ,ASφ)+,φ+ds+(ASφ,Sφ)S+(ReHφ,φ)S+i(ImHφ,φ)SφW21(S).

Let us prove the uniqueness of the solution. For this, consider homogenous boundary conditions: f=0,(νΓ,ASφ)=0 on ΓN and φ+=0 on ΓD. Then, (divSAS-H)φ=0 and Γ(νΓ,ASφ)+,φ+ds=0, and finally we get (47)(ASφ,Sφ)S+(ReHφ,φ)S=0,(ImHφ,φ)S=0.

Now let ReH(t)0, for all tS, and messuppReH0 (case (i)). Then from the first equality in (47), it follows that (48)(ASφ,Sφ)S=0,(ReHφ,φ)S=0. The first equality (ASφ,Sφ)S=0 yields φ(t)=C=const and by inserting this in the second one we get (49)0=(ReHφ,φ)S=CSH(t)dσ, and the conclusion φ(t)=C=const=0 is immediate.

If M=ImH(t)=const0 (case (ii)), the same conclusion follows from the second equality in (47): (50)0=(ImHφ,φ)S=M(φ,φ)S=MφL2(S)2, and, again, φ=0.

If ReH=0, either ImH(t)0, for all tS, or ImH(t)0, for all tS, and messuppImH0 (case (iii)); from equality (46) it follows that (51)(-(divSAS-H)φ,φ)S=(ASφ,Sφ)S+i(ImHφ,φ)S=0 and, consequently, if (divSAS-H)Sφ=0 for W21(S), then (52)(ASφ,Sφ)S=0,(ImHφ,φ)S=0. The conclusion φ=const=0 follows as in the case (i).

Therefore, Ker(divSAS-HI)={0}. Since the operator is self-adjoint, the same is true for the dual operator Coker(divSAS-HI)=Ker(divSAS-HI)={0} which, together with the Fredholm property of divSAS-HI:W1(S)W-1(S), yields the invertibility of this operator (of the operator in (44) for s=0 and p=2).

If s0 or p2, we proceed as follows. It is well known (see, e.g., ) that if an operator is Fredholm in the scale of Banach spaces (44), for all sR, 1<p<, and is invertible for only one pair of parameters (p,s)=(2,0), it is invertible for all values of the parameters sR, 1<p<.

To prove the second assertion (see [3, Lemma 5.2] for a particular case), we note that the natural domain of the operator divSAS is Xp,#s+1(S).

To prove that the image of the operator divSAS coincides with the space Xp,#s-1(S), let ψ0Xp,#s-1(S) be orthogonal to the image ImdivSAS of the operator. Then the equality (53)0=(divSASφ,ψ0)=(φ,divSASψ0) holds for all φXps+1(S). But then divSASψ0=0, which implies ψ0=const and, therefore, constants are only functions orthogonal to the image of the operator ImdivSAS. This proves that the image ImdivSAS:=divSASp coincides with the space Xp,#s-1(S).

Now, note that the operator -divSAS in the setting (54)-divSAS:W2,#1(S)W2,#-1(S) is positive definite (coercive). Indeed, due to (9), (31), and (43), we get (55)(-divSASφ,φ)=(ASφ,Sφ)CSφL2(S)2=CφW2,#1(S)2mmmmmmmmmmmmmφW2,#1(S). Therefore, divSAS has the trivial kernel Ker(divSAS)={0} in W2,#1(S) and is normally solvable (has the closed image). Since divSAS is self-adjoint and the spaces W2,#1(S) and W2,#-1(S) are adjoint (see (40)), the adjoint operator to (54) is normally solvable as well and has a trivial kernel.

The proof is accomplished as in the first part of the theorem.

Corollary 11 (cf. [<xref ref-type="bibr" rid="B6">3</xref>]).

For the operator divC(AC) on the open hypersurface C with the boundary C:=Γ, the following Green formulae are valid: (56)(divS(ACφ),ψ)C=(νΓ,(ACφ)+,ψ+)Γ-(ACφ,Cψ)C,(divS(ACφ),ψ)C-(φ,divS(ACψ))C=(νΓ,(ACφ)+,ψ+)Γ-(φ+,νΓ,(ACψ)+)Γ, where (φ,ψ)C denotes the scalar product of functions φ,ψC(C). We have encountered already the normal boundary derivative νΓ,(ACφ)+ in the mixed BVP (9).

Lemma 12 (see [<xref ref-type="bibr" rid="B12">18</xref>, <xref ref-type="bibr" rid="B14">19</xref>] (Lax-Milgram)).

Let B be a Banach space and let A(φ,ψ) be a continuous, bilinear form (57)A(·,·):B×BR and positive definite (58)A(φ,φ)CφB2φB,C>0. Let L(·):BR be a continuous linear functional.

A linear equation (59)A(φ,ψ)=L(ψ) has a unique solution φB for arbitrary ψB.

3. Mixed BVP for the Laplace-Beltrami Equation

Let again CS be a smooth subsurface of a closed hypersurface S and let Γ=C be its smooth boundary C=Γ. Let divCAC be the operator restricted to the hypersurface C. The boundary is divided into two parts C=Γ=ΓDΓN and in the present section we consider the boundary value problems for (8) with mixed boundary conditions in a weak classical formulation (4).

Note that functions  φWps(C) and φHps(C) have the trace φ+Wps-1/p(Γ) on the boundary, provided 1<p< and s>1/p (see  for details). Therefore if we look for a solution of the BVP (8) in the space W1(C), the trace u+ on ΓD exists and belongs to the space H1/2(ΓD).

Concerning the existence of the Neumann trace νΓ,AC+ in (8) for a solution uW1(C) is not guaranteed by the general trace theorem. But in this case, the first Green formula (56) ensures the existence of the Neumann trace. Indeed, by setting φ=u and inserting the data (divCAC)u(t)=f(t) from (8) into the first Green formula (56), we get the following: (60)(νΓ,(ACu)+,ψ+)Γ-(ACu,Cψ)C=(divC(ACu),ψ)C=(f,ψ)C, and finally we get (61)(νΓ,(ACu)+,ψ+)Γ=(f,ψ)C+(ACu,Cψ)C, for arbitrary ψW1(C). Since ψ+H1/2(Γ), the scalar product (ACu,Cψ)C in the right-hand side of equality (61) is correctly defined and defines correct duality in the left-hand side of the equality. Since ψ+H1/2(Γ) is arbitrary, by the duality argument, this gives that νΓ,(ACu)+ should be in the dual space, that is, in H-1/2(Γ).

To prove the forthcoming theorem about the unique solvability of the BVP (61) we need more properties of trace operators (called retractions) and their inverses, called coretractions (see [12, Section 2.7]).

To keep the exposition simpler we recall a very particular case of Lemma 4.8 from , which we apply in the present investigation.

Lemma 13 (see Lemma  4.8 in [<xref ref-type="bibr" rid="B5">1</xref>]).

Let s>0,sN, 1<p<, B(D) be a normal differential operator of the first order defined in the vicinity of the boundary Γ=rtialC and let A(D) be a normal differential operator of the second order defined on the surface C. Then there exists a continuous linear operator (62)B:Wps(Γ)Wps-1(Γ)Hps+1/p(C) such that (63)(BΦ)+=φ0,(B(D)BΦ)+=φ1,A(D)BΦH~ps-2+1/p(C), for arbitrary pair of functions Φ=(φ0,φ1), where φ0Wps(Γ) and φ1Wps-1(Γ).

Theorem 14.

The mixed boundary value problem (8) in the weak classical setting (4) and the additional constraint fH~0-1(Ω)H~-1(Ω) (see Remark 2), has a unique solution in the space W1(C).

Proof.

We commence by reduction of the BVP (8) to an equivalent one with the homogeneous Dirichlet condition. For this we extend the boundary data gW1/2(ΓD) up to some function g~W1/2(Γ) on the entire boundary Γ and apply Lemma 13: there exists a function GW1(C) such that G+(t)=g(t) for tΓD (actually G+=g~ almost everywhere on the boundary) and divC(ACG)W~-1(C¯).

For a new unknown function v:=u-G, we have the following equivalent reformulation of the BVP (8): (64)divC(ACv)(t)=f0(t),tC,v+(s)=0,onΓD,νΓ(s),(ACv)+(s)=h0(s),onΓN, where (65)f0:=f+divC(ACG)W~-1(C),h0:=h+νΓ,(ACG)+W-1/2(ΓN),v+W~1/2(ΓN). To justify the last inclusion v+W~1/2(ΓN), note that, due to our construction, the trace of a solution on ΓD vanishes v+ΓD=0.

Let Γ0Γ be a part of the boundary and, by Ws(Γ0,C), s>1/2, denote the space of functions φWs(Γ0,C) which has the trace on the boundary supported in Γ0; that is, φ+W~s-1/2(Γ0).

By inserting the data from the reformulated boundary value problem (64) into the first Green identity (56), where φ=ψ=v, we get (66)(ACv,Cv)C=(νΓ,(ACv)+,v+)ΓD+(νΓ,(ACv)+,v+)ΓN-(divC(ACv),v)C=(h0,v+)ΓN-(f0,v)C,mmmmmmmmmmmmmmmmmml.vW~1(ΓN,C). In the left-hand side of the equality (66) we have a symmetric bilinear form, which is positive definite (67)(ACv,Cv)CCCv2vW~1(ΓN,C), because A=[ajk]n×n is strictly positive definite matrix. On the other hand, all functions on the space W~1(ΓN,C) vanish on the part of the boundary ΓD, and Cv defines an equivalent norm on this space (68)CvvW1(ΓN)C1CvvW~1(ΓN,C). From (67) and (68) follows the positive definiteness: (69)(ACv,Cv)CC2vW1(ΓN)2vW~1(ΓN,C).

( h 0 , v + ) Γ N and (f0,v)C from equality (66) are correctly defined continuous functionals, because h0W-1/2(ΓN),f0W~-1(C), while their counterparts in the functional belong to the dual spaces v+W~1/2(ΓN) and vW~1(ΓN,C)W1(C).

The Lax-Milgram Lemma 12 accomplishes the proof.

Remark 15.

The Lax-Milgram Lemma 12 can be applied to the Dirichlet BVP (1), (4), but only after equivalent reformulation: due to Lemma 13 we can pick up a function GW1(C) such that G+=g and divC(ACG)W~-1(C).

For a new unknown function v:=u-G, we have the following equivalent reformulation of the BVP (5): (70)divC(ACv)(t)=f0(t),tC,v+(s)=0,onΓ, where (71)f0:=f+divC(ACG)W~-1(C),vW~1(C). By inserting the data from the reformulated boundary value problem (70) into the first Green identity (56), where φ=ψ=v, we get (72)(ACv,Cv)C=(νΓ,(ACv)+,v+)Γ-(divC(ACv),v)C=-(f0,v)C,mmmmmmmmmmmllvW~1(C). Now the Lax-Milgram Lemma can be applied, which proves the unique solvability of the Dirichlet BVP (1), (4).

Remark 16.

The Lax-Milgram Lemma 12 can be applied to the Neumann BVP (2), (4): (73)divC(ACv)(t)=f0(t),tC,νΓ,(ACv)+(s)=0,onΓ. By inserting the data from the boundary value problem (2) into the first Green identity (56), where φ=ψ=u, we get (74)(ACu,Cu)C=(νΓ,ACu+,u+)Γ-((divCAC)u,u)C=(h,u)Γ-(f,u)C,uW2,#1(C). We need the compatibility condition (6) to ensure that the equality in (74) will be fulfilled on the subspace W2,#1(C): if v=const, the left-hand side vanishes, while the right-hand side vanishes, (h,const)Γ-(f,const)C=0, if and only if the compatibility condition (6) holds: (75)(h,1)Γ-(f,1)C=Γh(s)ds-Cf(y)dσ=0, and the last representation in the equality is possible if h and f are regular integrable functions.

Due to (55), the bilinear form (ACv,Cv)C is positive definite and both functionals in the right-hand side of (74) are bounded on the subspace W2,#1(C). Now we can apply the Lax-Milgram Lemma and prove the existence of a unique solution vW2,#1(C) to the Neumann BVP (2), (4), provided the compatibility9 condition (6) holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first and the second authors were supported by the Grant of the Shota Rustaveli Georgian National Science Foundation GNSF/DI/10/5-101/12 and the second author was also supported by the Grant of the Shota Rustaveli Georgian National Science Foundation GNSF/PG/76/5-101/13.

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