Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form

Abstract

We study the Dirichlet problem for second-order elliptic operators in double divergence form, which arise as formal adjoints of non-divergence form operators and include the stationary Fokker-Planck-Kolmogorov equation. Assuming that the leading coefficients have Dini mean oscillation and that the lower-order coefficients satisfy natural integrability conditions, we construct the Perron solution in arbitrary bounded domains. We prove that a boundary point is regular with respect to the operator if and only if it satisfies the classical Wiener criterion for the Laplacian. In particular, the Dirichlet problem is uniquely solvable in every bounded domain that is regular for the Laplacian.

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