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

Svitlana Mayboroda, Vladimir Maz'Ya

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)287-334
Number of pages48
JournalInventiones Mathematicae
Volume175
Issue number2
DOIs
StatePublished - Feb 1 2009

Fingerprint Dive into the research topics of 'Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation'. Together they form a unique fingerprint.

Cite this