Moscow Mathematical Journal

Volume 26, Issue 3, July–September 2026 pp. 279–301.

The Dirichlet Problem for Double Divergence Form Elliptic Equations with Measures as Boundary Conditions

Authors: V. I. Bogachev (1) and S. V. Shaposhnikov (2)
Author institution:(1) Department of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia
(2) National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russia

Summary:

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.

2020 Math. Subj. Class. 35J15, 35J25, 35J67.

Keywords: Double divergence form elliptic equation, Kolmogorov equation, Dirichlet problem, boundary condition.

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