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Mixed Boundary Value Problem on Hypersurfaces
The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation on a smooth hypersurface with the boundary in . is an bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts and on the Dirichlet boundary conditions are prescribed, while on 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 is proved, which is interpreted as the invertibility of this operator in the setting , where is a subspace of the Bessel potential space and consists of functions with mean value zero.
Let be a smooth subsurface of a closed hypersurface in the Euclidean space (see Section 2 for details) and let be its smooth boundary . Let , be Günter’s tangential derivatives, and let be the Laplace-Beltrami operator restricted to the hypersurface (see [1–3] and Section 2 below for details).
In , the boundary value problem for the Laplace-Beltrami equation with the Dirichlet boundary condition and with the Neumann boundary condition were considered where is the unit normal vector field to the boundary and tangent to the hypersurface and denotes the trace on the boundary. The derivative is tangent to the hypersurface and normal with respect to the boundary .
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 should be fulfilled, which guarantees the existence and the uniqueness of solution.
If and are regular integrable functions, the compatibility condition (6) acquires the form
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 , 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: where is a decomposition of the boundary into two connected parts, is an strictly positive definite matrix, and for all . 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 which is the orthogonal complement to the subspace of those distributions from which are supported on the boundary of the domain only.
For the non-classical setting (5) we schould impose a similar constraint , which is defined as in the case .
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 in has the natural structure of an -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 such as curvature. Inherently, these PDEs are originally written in local coordinates, intrinsic to the manifold structure of .
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 lies in the existence of fundamental solution: in 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 , and prove that it is an invertible operator. Another description of the space is that it consists of all functions (distributions if , which have the zero mean value, ). 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 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 of the Euclidean space is called a hypersurface if it has a covering and coordinate mappings
such that the corresponding differentials
have the full rank
that is, all points of are regular for for all .
Such a mapping is called an immersion as well.
Here and in what follows refers to the matrix with the listed vectors as columns.
A hypersurface is called smooth if the corresponding coordinate diffeomorphisms in (10) are smooth (-smooth). Similarly is defined a -smooth hypersurface.
The next definition of a hypersurface is implicit.
Definition 4. Let and be a compact domain. An implicit -smooth hypersurface in is defined as the set where is a -mapping, which is regular .
Stoke’s derivatives are concrete examples of tangential operators
Corollary 5 (Green’s formula; cf. ). Let be a domain with Lipschitz boundary. For the Laplace operator and functions , the following I and II Green formulae are valid: Integrals in (16) are understood in the sense of duality between spaces and , and , and so forth.
Let be a closed hypersurface in and let be a smooth subsurface of , given by an immersion with a boundary , given by another immersion and let be the outer unit normal vector field to and let denote an extended unit field in a neighborhood of . is the outer normal vector field to the boundary , which is tangential to .
A curve on a smooth surface is a mapping of a line interval to .
A vector field on a domain in is a mapping where and is the element of the natural Cartesian basis in : in the Euclidean space .
By we denote the set of all smooth vector fields on .
A vector field defines the first order differential operator where 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 , and its principal symbol is given by the matrix-valued function
To distinguish an open and a closed hypersurface, we use the notation for a closed hypersurface without the boundary (we remind the reader that the notation is reserved for an open hypersurface with the boundary ).
Definition 6. We say that is a tangential operator to the hypersurface , with unit normal , if
Lemma 7. Let be, as in (23), a first order differential operator with -smooth coefficients. is tangential if and only if the adjoint operator is tangential.
If is tangential to and is defined in a neighborhood of , then for every function defined in a neighborhood of .
We continue with the definition of the surface divergence , the surface gradient , and the surface Laplace-Beltrami operator .
According to the classical differential geometry, the surface gradient of a function is defined by and the surface divergence of a smooth tangential vector field is defined by where denotes the Christoffel symbols and is the covariant Riemann metric tensor, while is the inverse to it-the contravariant Riemann tensor.
is the negative dual to the surface gradient: The Laplace-Beltrami operator on is defined as the composition
Theorem 8 (cf. ). For any function , one has Also, for a 1-smooth tangential vector field , The Laplace-Beltrami operator on takes the form
Corollary 9 (cf. ). Let be a smooth closed hypersurface. The homogeneous equation has only a constant solution in the space .
Proof. Due to (31) and (35), we get which gives . But the trivial surface gradient means constant function (this is easy to ascertain by analysing the definition of Günter’s derivatives; see, e.g., ).
Let be a nontrivial , smooth closed or open hypersurface, , and . For the definitions of Bessel potential and Sobolev-Slobodeckii spaces for a closed smooth manifold , we refer to  (also see [1, 13, 14]). For the Sobolev-Slobodetski and Bessel potential spaces coincide (i.e., the norms are equivalent).
Let be a subsurface of a smooth closed surface , , with the smooth boundary . The space is defined as the subspace of those functions , which are supported in the closure of the subsurface, , whereas denotes the quotient space , and is the complementary subsurface to . The space can be identified with the space of distributions on which have an extension to a distribution . Therefore, , where denotes the restriction operator of functions (distributions) from the surface to the subsurface .
By we denote one of the spaces: and Sobolev-Slobodecki (if is closed or open), and by denote one of the spaces: and (if is open). Consider the space It is obvious that does not contain constants: if , then and . Moreover, decomposes into the direct sum and the dual (adjoint) space is In fact, the decomposition (39) follows from the representation of arbitrary function , because the average of the difference of a function and its average is zero: .
Since the Sobolev space with integer smoothness parameter 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 the equivalent norm is
The description (40) of the dual space follows from the fact that the dual space to is (see ) and, therefore, due to the decomposition (39) and Hahn-Banach theorem the dual space to should be embedded into . The only functional from that vanishes on the entire space is constant (see definition (37)). After detaching this functional the remainder coincides, due to (39), with the space , which is the dual to .
Theorem 10. Let be -smooth and . Let be the same as in (37)–(40).
Let have one of the following properties: (i) has a nonnegative real part , for all , and ;(ii) has a constant complex part ;(iii), , and the complex part does not change the sign: either , for all , or , for all .
The perturbed operator is invertible, which can be interpreted as the existence of the fundamental solution to .
The operator itself is invertible between the spaces with detached constants (see (37)) And, therefore, 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 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 in (44) is Fredholm, for all and (it has a parametrix if 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: on and on . Then, and , and finally we get
Now let , for all , and (case (i)). Then from the first equality in (47), it follows that The first equality yields and by inserting this in the second one we get and the conclusion is immediate.
If (case (ii)), the same conclusion follows from the second equality in (47): and, again, .
If , either , for all , or , for all , and (case (iii)); from equality (46) it follows that and, consequently, if for , then The conclusion follows as in the case (i).
Therefore, . Since the operator is self-adjoint, the same is true for the dual operator which, together with the Fredholm property of , yields the invertibility of this operator (of the operator in (44) for and ).
If or , 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 , , and is invertible for only one pair of parameters , it is invertible for all values of the parameters , .
To prove the second assertion (see [3, Lemma 5.2] for a particular case), we note that the natural domain of the operator is .
To prove that the image of the operator coincides with the space , let be orthogonal to the image of the operator. Then the equality holds for all . But then , which implies and, therefore, constants are only functions orthogonal to the image of the operator . This proves that the image coincides with the space .
Now, note that the operator in the setting is positive definite (coercive). Indeed, due to (9), (31), and (43), we get Therefore, has the trivial kernel in and is normally solvable (has the closed image). Since is self-adjoint and the spaces and 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. ). For the operator on the open hypersurface with the boundary , the following Green formulae are valid: where denotes the scalar product of functions . We have encountered already the normal boundary derivative in the mixed BVP (9).
Lemma 12 (see [18, 19] (Lax-Milgram)). Let be a Banach space and let be a continuous, bilinear form
and positive definite
Let be a continuous linear functional.
A linear equation has a unique solution for arbitrary .
3. Mixed BVP for the Laplace-Beltrami Equation
Let again be a smooth subsurface of a closed hypersurface and let be its smooth boundary . Let be the operator restricted to the hypersurface . The boundary is divided into two parts 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 and have the trace on the boundary, provided and (see  for details). Therefore if we look for a solution of the BVP (8) in the space , the trace on exists and belongs to the space .
Concerning the existence of the Neumann trace in (8) for a solution 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 and inserting the data from (8) into the first Green formula (56), we get the following: and finally we get for arbitrary . Since , the scalar product in the right-hand side of equality (61) is correctly defined and defines correct duality in the left-hand side of the equality. Since is arbitrary, by the duality argument, this gives that should be in the dual space, that is, in .
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 ). Let , , be a normal differential operator of the first order defined in the vicinity of the boundary and let be a normal differential operator of the second order defined on the surface . Then there exists a continuous linear operator such that for arbitrary pair of functions , where and .
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 up to some function on the entire boundary and apply Lemma 13: there exists a function such that for (actually almost everywhere on the boundary) and .
For a new unknown function , we have the following equivalent reformulation of the BVP (8): where To justify the last inclusion , note that, due to our construction, the trace of a solution on vanishes .
Let be a part of the boundary and, by , , denote the space of functions which has the trace on the boundary supported in ; that is, .
By inserting the data from the reformulated boundary value problem (64) into the first Green identity (56), where , we get In the left-hand side of the equality (66) we have a symmetric bilinear form, which is positive definite because is strictly positive definite matrix. On the other hand, all functions on the space vanish on the part of the boundary , and defines an equivalent norm on this space From (67) and (68) follows the positive definiteness:
and from equality (66) are correctly defined continuous functionals, because , while their counterparts in the functional belong to the dual spaces and .
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 such that and .
For a new unknown function , we have the following equivalent reformulation of the BVP (5): where By inserting the data from the reformulated boundary value problem (70) into the first Green identity (56), where , we get 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): By inserting the data from the boundary value problem (2) into the first Green identity (56), where , we get We need the compatibility condition (6) to ensure that the equality in (74) will be fulfilled on the subspace : if , the left-hand side vanishes, while the right-hand side vanishes, , if and only if the compatibility condition (6) holds: and the last representation in the equality is possible if and are regular integrable functions.
Due to (55), the bilinear form is positive definite and both functionals in the right-hand side of (74) are bounded on the subspace . Now we can apply the Lax-Milgram Lemma and prove the existence of a unique solution 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.
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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