The Robin Mean Value Equation I: A Random Walk Approach to the Third Boundary Value Problem

作     者：Lewicka, Marta Peres, Yuval 

作者机构：Univ Pittsburgh Dept Math 139 Univ Pl Pittsburgh PA 15260 USA 

出 版 物：《POTENTIAL ANALYSIS》 (位势分析)

年 卷 期：2023年第59卷第4期

页      面：1695-1726页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：NSF [DMS-1613153] 

主　　题：Robin problem Third boundary value problem Oblique boundary conditions Dynamic programming principle Random walk Finite difference approximations Viscosity solutions 

摘      要：We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on C-1,C-1 -regular domains, we prove existence, uniqueness, equiboundedness and the comparison principle for solutions to (RMV). For the continuous right hand side of (RL), we show that solutions to (RMV) converge uniformly, in the limit of the vanishing radius of averaging, to the unique W-2,W- P solution, which coincides with the unique viscosity solution of (RL). We further prove the lower bound on solutions to (RMV), which is consistent with the optimal lower bound for solutions to (RL). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks.