Polyharmonic capacity and Wiener test of higher order

TL;DR

This paper establishes a Wiener test for boundary regularity of solutions to the polyharmonic operator, introducing a new notion of capacity and providing necessary and sufficient conditions for regularity.

Contribution

It introduces a new polyharmonic capacity and proves the Wiener test for boundary regularity for arbitrary order polyharmonic equations.

Findings

Introduced a new notion of polyharmonic capacity.

Established Wiener test for boundary regularity.

Provided necessary and sufficient capacity conditions.

Abstract

In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by [Mayboroda, Maz'ya, Invent. Math. 2009]. As a preliminary stage of this work, in [Mayboroda, Maz'ya, Invent. Math. 2013] we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order.