2004/08/19 by Rodrigo Bañuelos, Rodrigo Banuelos, Banuelos, Rodrigo +2
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Probability (math.PR) #math.AP #math.PR
paper · pdf · doi:10.48550/arxiv.math/0408267
openalex publication_date 2004/08/19 · arxiv created 2004/08/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the "mixed Steklov problem," was established in a previous paper of the authors. From this, a variational characterization for the eigenvalues λn, n≥ 1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ1. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ_* - λ1 where λ_* is the eigenvalue corresponding to the "first" antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ_* - λ1 and on a weighted Poincaré--type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0