Mixed Boundary Value Problem on Hypersurfaces

and on Γ D the Dirichlet boundary conditions are prescribed, while on Γ N 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(A∇S) is proved, which is interpreted as the invertibility of this operator in the setting H s p,#(S) → H s−2 p,# (S), where H s p,#(S) is a subspace of the Bessel potential space and consists of functions with mean value zero.

In [3], the boundary value problem for the Laplace-Beltrami equation with the Dirichlet boundary condition were considered where ^Γ := (] Γ,1 , . . ., ] Γ, ) ⊤ is the unit normal vector field to the boundary Γ and tangent to the hypersurface C and  + denotes the trace on the boundary.The derivative is tangent to the hypersurface C and normal with respect to the boundary Γ.
The BVPs (1) and ( 2) were investigated in [3] in the following classical weak setting: and also in nonclassical weak setting: and the following was proved.
For the solvability of the Neumann problems (2), ( 4) and (2), (5), the necessary and sufficient compatibility condition International Journal of Differential Equations should be fulfilled, which guarantees the existence and the uniqueness of solution.
If  and ℎ are regular integrable functions, the compatibility condition (6) acquires the form 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 [3] 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. [2] and applies potential method.Similar problems, for  = 2, by different technique were investigated earlier in the paper of Mitrea and Taylor [4].
The purpose of the present paper is to investigate the boundary value problems for the anisotropic Laplace equation with mixed boundary conditions: where C = Γ = Γ  ∪ Γ  is a decomposition of the boundary into two connected parts,  = {  } is an  ×  strictly positive definite matrix, and for all  ∈ C. 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 [14], 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−1 0 (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 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 [5]).
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 [10] for the Navier-Stokes equations on spherical domains, as well as the references therein.
A hypersurface S in R  has the natural structure of an (−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 R  lies in the existence of fundamental solution: in R  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 , for all 1 <  < ∞,  ∈ R, and prove that it is an invertible operator.Another description of the space W  ,# (S) is that it consists of all functions  ∈ W   (S) (distributions if  < 0, which have the zero mean value, (, 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 W  ,# (S) → W −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).

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 S ⊂ R  of the Euclidean space is called a hypersurface if it has a covering S = ⋃  =1 S  and coordinate mappings such that the corresponding differentials have the full rank rank Θ  () =  − 1, ∀ ∈   ,  = 1, . . ., ,  = 1, . . ., ; that is, all points of   are regular for Θ  for all  = 1, . . ., .Such a mapping is called an immersion as well.
A hypersurface is called smooth if the corresponding coordinate diffeomorphisms Θ  in (10) are smooth ( ∞smooth).Similarly is defined a -smooth hypersurface.
The next definition of a hypersurface is implicit.
Definition 4. Let  ⩾ 1 and  ⊂ R  be a compact domain.An implicit   -smooth hypersurface in R  is defined as the set where
Let S be a closed hypersurface in R  and let C be a smooth subsurface of S, given by an immersion with a boundary Γ = C, given by another immersion and let ^(X) be the outer unit normal vector field to C and let N() denote an extended unit field in a neighborhood  C of C. ^Γ() 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 of a line interval I to C.
A vector field U ∈ V(Ω) defines the first order differential operator where F  U () is the orbit of the vector field .Let be a first order differential operator with real valued (variable) matrix coefficients, acting on vector-valued functions in R  , and its principal symbol is given by the matrix-valued function To distinguish an open and a closed hypersurface, we use the notation S for a closed hypersurface without the boundary rtialS = 0 (we remind the reader that the notation C is reserved for an open hypersurface with the boundary Γ := rtialS).Definition 6.We say that  is a tangential operator to the hypersurface S, with unit normal ^, if  (; ^) = 0 on the hypersurface S. (25) for every  1 function  defined in a neighborhood of S.
We continue with the definition of the surface divergence div S , the surface gradient ∇ S , and the surface Laplace-Beltrami operator Δ S .
According to the classical differential geometry, the surface gradient ∇ S of a function  ∈  1 (S) is defined by and the surface divergence of a smooth tangential vector field V is defined by where Γ   denotes the Christoffel symbols and  := [  ] is the covariant Riemann metric tensor, while  −1 := [  ] is the inverse to it-the contravariant Riemann tensor.div S is the negative dual to the surface gradient: The Laplace-Beltrami operator Δ S on S is defined as the composition Theorem 8 (cf.[2]).For any function  ∈  1 (S), one has Also, for a 1-smooth tangential vector field The Laplace-Beltrami operator Δ S on S takes the form M 2  , ∀ ∈  2 (S) . (34) Corollary 9 (cf.[2]).Let S be a smooth closed hypersurface.The homogeneous equation has only a constant solution in the space W 1 (S).
Proof.Due to (31) and ( 35), we get which gives ∇ S  = 0.But the trivial surface gradient means constant function  = const (this is easy to ascertain by analysing the definition of Günter's derivatives; see, e.g., [3]).
Let C be a subsurface of a smooth closed surface M, C ⊂ M, with the smooth boundary Γ := rtialS.The space H  (C) is defined as the subspace of those functions  ∈ H   (M), which are supported in the closure of the subsurface, supp  ⊂ C, whereas H   (C) denotes the quotient space H   (C) = H   (M)/ H  (C  ), and C  := M \ C is the complementary subsurface to C. The space H   (C) can be identified with the space of distributions  on C which have an extension to a distribution ℓ ∈ H   (M).Therefore,  C H   (M) = H   (C), where  C denotes the restriction operator of functions (distributions) from the surface M to the subsurface R  .
By X   (M) we denote one of the spaces: It is obvious that and  0 = 0.Moreover, X   (M) decomposes into the direct sum and the dual (adjoint) space is In fact, the decomposition (39) follows from the representation of arbitrary function  ∈ X   (M), because the average of the difference of a function and its average is zero: ( 0 ) aver = ( −  aver ) aver = 0.
Since the Sobolev space W  ,# (M) with integer smoothness parameter  = 1, 2, . . .does not contain constants, due to Corollary 9 the equivalent norm in this space can also be defined as follows: In particular, in the space W 1 ,# (M) the equivalent norm is The description (40) of the dual space follows from the fact that the dual space to X   (M) is X −   (M) (see [12]) and, therefore, due to the decomposition (39) and Hahn-Banach theorem the dual space to X  ,# (M) should be embedded into X −   (M).The only functional from X −   (M) that vanishes on the entire space X  ,# (M) is constant 1 ∈ X −   (M) (see definition (37)).After detaching this functional the remainder coincides, due to (39), with the space X −   ,# (M), which is the dual to X  ,# (M).

The perturbed operator
is invertible, which can be interpreted as the existence of the fundamental solution to div S ∇ S − H.
And, therefore, div S ∇ S 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 W 1 (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 div S ∇ S − H in (44) is Fredholm, for all  ∈ R and 1 <  < ∞ (it has a parametrix if S is infinitely smooth; see [13,15,16]).On the other hand, ( Let us prove the uniqueness of the solution.For this, consider homogenous boundary conditions:  = 0, (^Γ, ∇ S ) = 0 on Γ  and  + = 0 on Γ  .Then, (div S ∇ S − H) = 0 and ∫ Γ ⟨(^Γ, ∇ S ) + ,  + ⟩ = 0, and finally we get The conclusion  = const = 0 follows as in the case (i).Therefore, Ker(div S ∇ S − H) = {0}.Since the operator is self-adjoint, the same is true for the dual operator Coker(div S ∇ S − H) = Ker(div S ∇ S − H) = {0} which, together with the Fredholm property of
Let  be, as in (23), a first order differential operator with  1 -smooth coefficients.is tangential if and only if the adjoint  * operator is tangential.If  is tangential to S and  is defined in a neighborhood of S, then