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

• 成果类型: Article
• 署名作者: Mayboroda, Svitlana; Maz'ya, Vladimir
• 署名单位: Brown University; University System of Ohio; Ohio State University; University of Liverpool; Linkoping University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0150-x
• 发表日期: 2009
• 页码: 287-334
• 关键词: elliptic-equations; lipschitz-domains; criterion; Operators; BOUNDARY

摘要

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.