An optimal anisotropic Poincaré inequality for convex domains

Guofang Wang and Chao Xia

Vol. 258 (2012), No. 2, 305–326

DOI: 10.2140/pjm.2012.258.305

Abstract

In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of the anisotropic Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincaré inequality for convex domains, which generalizes the classical result of Payne and Weinberger. A lower bound of the first (nonzero) eigenvalue of the anisotropic Laplacian with the Dirichlet boundary condition is also proved.

Keywords

anisotropic Laplacian, first eigenvalue, gradient estimate, optimal Poincaré inequality

Mathematical Subject Classification 2010

Primary: 35P15

Secondary: 35J62, 35P30

Milestones

Received: 4 October 2011

Revised: 26 January 2012

Accepted: 22 May 2012

Published: 1 August 2012