Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation

TL;DR

This paper develops new integral identities to analyze solutions of the biharmonic equation in arbitrary domains, establishing gradient boundedness, Green function estimates, and Wiener-type criteria for solution regularity.

Contribution

Introduction of novel integral identities enabling analysis of biharmonic solutions in general domains, including boundedness and regularity criteria.

Findings

Gradient of solutions is bounded in any three-dimensional domain.

Pointwise estimates for derivatives of the biharmonic Green function.

Wiener-type conditions characterize the continuity of the solution's gradient.

Abstract

The behavior of solutions to the biharmonic equation is well-understood in smooth domains. In the past two decades substantial progress has also been made for the polyhedral domains and domains with Lipschitz boundaries. However, very little is known about higher order elliptic equations in the general setting. In this paper we introduce new integral identities that allow to investigate the solutions to the biharmonic equation in an arbitrary domain. We establish: (1) boundedness of the gradient of a solution in any three-dimensional domain; (2) pointwise estimates on the derivatives of the biharmonic Green function; (3) Wiener-type necessary and sufficient conditions for continuity of the gradient of a solution.