The Dirichlet problem for double divergence form elliptic equations with measures as boundary conditions

TL;DR

This paper investigates the Dirichlet problem for elliptic equations in double divergence form with low-regularity coefficients and measure boundary conditions, establishing broad solvability results and applications to Fokker-Planck-Kolmogorov equations.

Contribution

It introduces a framework for solving double divergence form elliptic equations with measure boundary conditions under broad assumptions, extending existing theory.

Findings

Established solvability of the Dirichlet problem under broad conditions

Showed solutions on a domain also solve subdomain problems

Applied results to stationary Fokker-Planck-Kolmogorov equations

Abstract

We introduce and study the Dirichlet problem for double divergence form elliptic equations with coefficients of low regularity and boundary conditions given by general Borel measures. Under broad assumptions we establish the solvability of this problem. It is also shown that a solution to a double divergence form equation on a domain serves as a solution to the Dirichlet problem on inner subdomains. The obtained results are applied to the study of properties of solutions to stationary Fokker--Planck--Kolmogorov equations.