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The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation

Let

In [

The BVPs (

The Dirichlet problems (

For the solvability of the Neumann problems (

If

In Remarks

The investigation in [

The purpose of the present paper is to investigate the boundary value problems for the anisotropic Laplace equation with mixed boundary conditions:

The nonclassical weak setting (

As shown in [

For the non-classical setting (

For the classical setting (

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, [

A hypersurface

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

The structure of the paper is as follows. In Section

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 [

A subset

Such a mapping is called an

Here and in what follows

A hypersurface is called

The next definition of a hypersurface is

Let

Let

Let

A

A

By

A vector field

Let

To distinguish an open and a closed hypersurface, we use the notation

We say that

Let

If

We continue with the definition of the surface divergence

According to the classical differential geometry, the

For any function

Let

Due to (

Let

Let

The spaces

By

Since the Sobolev space

The description (

Let

Let

The perturbed operator

The operator

The first part of the theorem is proved in [

First of all, note that the operator (

Let us prove the uniqueness of the solution. For this, consider homogenous boundary conditions:

Now let

If

If

Therefore,

If

To prove the second assertion (see [

To prove that the image of the operator

Now, note that the operator

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

For the operator

Let

A linear equation

Let again

Note that functions

Concerning the existence of the Neumann trace

To prove the forthcoming theorem about the unique solvability of the BVP (

To keep the exposition simpler we recall a very particular case of Lemma 4.8 from [

Let

The mixed boundary value problem (

We commence by reduction of the BVP (

For a new unknown function

Let

By inserting the data from the reformulated boundary value problem (

The Lax-Milgram Lemma

The Lax-Milgram Lemma

For a new unknown function

The Lax-Milgram Lemma

Due to (

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.