Functional inequalities for Brownian motion on manifolds with sticky-reflecting boundary diffusion

• 成果类型: Article; Early Access
• 署名作者: Bormann, Marie; von Renesse, Max; Wang, Feng-Yu
• 署名单位: Leipzig University; Max Planck Society; Tianjin University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01349-2
• 发表日期: 2024
• 关键词: sobolev trace inequalities; spectral asymptotics; EIGENVALUE; laplacian; constant; EQUATIONS; balls; WORST

摘要

We prove geometric upper bounds for the Poincar & eacute; and Logarithmic Sobolev constants for Brownian motion on manifolds with sticky reflecting boundary diffusion i.e. extended Wentzell-type boundary condition under general curvature assumptions on the manifold and its boundary. The method is based on an interpolation involving energy interactions between the boundary and the interior of the manifold. As side results we obtain explicit geometric bounds on the first nontrivial Steklov eigenvalue, for the norm of the boundary trace operator on Sobolev functions, and on the boundary trace logarithmic Sobolev constant. The case of Brownian motion with pure sticky reflection is also treated.