The Wiener test for higher order elliptic equations

Abstract

Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear divergence and nondivergence equations with discontinuous coefficients, equations with degenerate quadratic form, quasilinear and fully nonlinear equations, as well as equations on Riemannian manifolds, graphs, groups, and metric spaces. A common feature of these equations is that all of them are of second order, and Wiener type characterizations for higher order equations have been unknown so far. Indeed, the increase of the order results in the loss of the maximum principle, Harnack's inequality, barrier techniques, and level truncation arguments, which are ingredients in different proofs related to the Wiener test for the second order equations. In the present work we extend Wiener's result to elliptic differential operators $L(\partial)$ of order $2m$ in the Euclidean space ${\bf R}^n$ with constant real coefficients $$L(\partial)=(-1)^m\sum_{|α|=|β|=m}a_{αβ} \partial^{α+ β}.$$ The results can be extended to equations with variable (for example, Hölder continuous) coefficients in divergence form but we leave aside this generalization to make exposition more lucid.