On the Convergence of Solutions for SPDEs under Perturbation of the Domain

Zhongkai Guo,Jicheng Liu,Wenya Wang

doi:10.1155/2016/5425639

Abstract

We investigate the effect of domain perturbation on the behavior of mild solutions for a class of semilinear stochastic partial differential equations subject to the Dirichlet boundary condition. Under some assumptions, we obtain an estimate for the mild solutions under changes of the domain.

Highlights

Domain perturbation, or sometimes referred to as “perturbation of the boundary,” for boundary value problems is a special topic in perturbation problems
We show how the mild solution of the stochastic differential equations behaves when domain Ωε converges to domain Ω under a certain sense
This paper is organized as follows: In Section 2, we review the results of the existence and uniqueness to the stochastic partial differential equation which we consider

Summary

Introduction

Sometimes referred to as “perturbation of the boundary,” for boundary value problems is a special topic in perturbation problems. The work of [2] obtains the convergence of solution for elliptic equation subject to Dirichlet boundary condition; necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators. We recommend [9], caring about the coefficients perturbation for semilinear stochastic partial differential equations; as we mention above it belongs to smooth domain perturbation problem. Under the condition of the operator norm convergence of resolvent operators, the author of [11] gives a distance estimate of the inertial manifolds for partial differential equations of evolutionary type under perturbation of the domain. As there are not many results on domain perturbation with noise in the dynamics, inspired by [11], we take similar conditions as in [11] to consider the convergence of solution for stochastic partial differential equations under perturbation of the domain. We will write the dependence of constant on parameters explicitly if it is essential

Preliminaries

Solution under Perturbation of the Domain