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

Vladyslav Shram

Abstract


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 Laplaces 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.

Keywords


Measure; Divergence; Laplacian; Laplace’s equation; Dirichlet problem

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References


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