Dirichlet problem in a ball for Laplace’s equation with Laplacian with respect to a measure

In this article, we study a Dirichlet problem for a generalized Laplace’s equation. We consider a construction of Laplacian with respect to a measure, that generalizes the classical Laplace’s operator to the case of an arbitrary measure. Certain properties of the constructed Laplacian are studied and a Dirichlet problem for Laplace’s equation with this new Laplacian is set. We propose a general solution construction framework for the Dirichlet problem in a ball in 2- and 3-dimensional spaces in the case of densities, that are invariant to orthogonal transforms. Using this framework we find explicit solutions for several important and rich families of densities, with the Gaussian density among them.


Introduction
The construction of divergence with respect to (w.r.t.) a measure and Laplacian w.r.t. a measure allows us to generalize classical divergence and Laplacian operators to the case of non-invariant measure. The problem of generalizing classical results of mathematical physics to the case of non-invariant measure is quite promising from the standpoint of the possibility of transferring certain results to the case of an infinitedimensional argument. For more detail, see the following papers: (Bogdanskii, 2012), (Bogdanskii, 2013), (Bogdanskii & Sanzharevskii, 2014), (Bogdanskii & Potapenko, 2016) and (Bogdanskii & Potapenko, 2017).
Let us now describe the construction of divergence w.r.t. a measure and Laplacian w.r.t. a measure we will be using in this paper.
Let ( , A, ) be a space with a measure, where = R , A is a Borel -algebra of subsets of , and is a signed measure (from now on -just «measure») on A (whether finite or infinite).
Let us also consider ∈ 1 ( , ), where 1 ( , ) denotes a space of all vector fields on with values in that are continuously differentiable, and are bounded on together with their first derivatives. By Φ ( 0 ) we denote the flow of the vector field at the time valued at the point 0 ∈ .
Thus, for every fixed ∈ R we've got a map R ∋ 0 ↦ → Φ ( 0 ) ∈ R . One can prove that it is a diffeomorphism.
Considering as a parameter, we obtain a one-parameter family of diffeomorphisms Φ . This family is called a «flow of vector field ».
Since for every ∈ R: Φ is a diffeomorphism, one can easily prove next Proposition 1.1. If ∈ A, then for each ∈ R : Φ ( ) ∈ A. Furthermore, for every ∈ R a map A ∋ ↦ → (Φ ( )) is a measure.
We will also need the next theorem.
Proof. See (Bogachev, 2007 Remark 1.4. The fact that is a measure immediatly follows from Nikodym-Vitali theorem. It turns out that the derivative is absolutely continuous w.r.t. (see for example (Bogachev, 2010)). Its density Let's now consider a measure that has a density w.r.t. some measure , which is differentiable along a field (we denote such a measure by · ). It turns out that in such a case we can rewrite div · in terms of div .
Proposition 1.5. Let : → R; ∈ 1 ( ), where 1 ( ) denotes a space of all continuously differentiable real-valued functions on that are bounded on together with their first derivative. If measure is differentiable along then measure · is differentiable along and the next equality holds Proof. See (Bogdanskii & Sanzharevskii, 2014).
It is also known (and one can easily prove it) that if a field ∈ 1 ( ) then the Lebesgue measure («volume») is differentiable along and, furthermore, div = div = ∑︀ =1 . Definition 1.6. Laplace's operator w.r.t. a measure is defined as follows Remark 1.7. From the definition 1.6 we see that Laplace's operator w.r.t. a measure is well-defined only for those functions ∈ 2 ( ) for which the field # » grad is bounded on and is differentiable along be an open ball with the radius and the center at ⃗ 0 ∈ R . Let be the Lebesgue measure on R .
We now consider the next Dirichlet problem. Find a function : where ℎ is some predefined continuous function on border of the ball. The uniqueness of the solution of the problem (2) immediately follows from the maximum principle for Laplacian w.r.t. a measure (see (Bogdanskii, 2016)).
According to proposition 1.5 we can rewrite problem (2) in the next form

Solutions Construction Framework
For 2-and 3-dimentional cases there were obtained a general framework for constructing solutions of the Dirichlet problem (3). With its help all the complexity of solving a Dirichlet problem in these cases can be reduced to solving an ordinary differential equation of the special type. So, let's describe these framework (or «recipe» for constructing solutions) in more detail.

2-dimensional case
First, we have to solve next ordinary differential equation

3-dimensional case
Similarly to 2-dimensional case, first we have to solve ordinary differential equation of the next form (5) Analogically, for each ∈ N ∪ {0} we have to find a solution ( ) of equation (5), that is bounded on the segment [0, ] together with its derivative and such that Similarly, we have the next result.

Explicit Solutions
Using the framework, described above, there were obtained explicit solutions of the Dirichlet problem in a ball for some important special cases of densities .

Summary
In this article the Dirichlet problem for Laplace's equation with Laplacian of a special form was studied. Certain important properties of the new Laplacian were presented, which helped to rewrite our Dirichlet problem in the form of the classical mathematical physics problem. We presented the general framework (or scheme) for constructing solutions for the Dirichlet problem in a ball in 2-and 3-dimensional spaces in the case of densities, which are invariant to orthogonal transforms. Then using this framework we obtained explicit solutions for this Dirichlet problem for several important and rich families of densities, one of which includes, among others, the Gaussian density.