Stochastic pseudomonotone parabolic obstacle problem: well-posedness $\&$ Lewy-Stampacchia's inequalities

TL;DR

This paper establishes the well-posedness of stochastic obstacle problems involving nonlinear pseudomonotone operators with multiplicative noise, and explores variational inequalities and Lewy-Stampacchia's inequalities in this context.

Contribution

It introduces a novel framework for analyzing stochastic obstacle problems with pseudomonotone operators, proving existence, uniqueness, and strong solutions.

Findings

Existence of martingale solutions via Prokhorov and Skorokhod theorems.

Path-wise uniqueness established through L1-contraction.

Foundation for variational inequalities and Lewy-Stampacchia's inequalities in stochastic setting.

Abstract

We consider obstacle problems for nonlinear stochastic evolution equations. More precisely, the leading operator in our equation is a nonlinear, second order pseudomonotone operator of Leray-Lions type. The multiplicative noise term is given by a stochastic integral with respect to a Q-Wiener process. We show well-posedness of the associated initial value problem for random initial data on a bounded domain with a homogeneous Dirichlet boundary condition. First, we consider a singular perturbation of our problem by a higher order operator. Through the a priori estimates for the approximate solutions of the singular perturbation, only weak convergence is obtained. This convergence is not compatible with the nonlinearities in the equation. Therefore we use the theorems of Prokhorov and Skorokhod to establish existence of martingale solutions. Then, path-wise uniqueness follows from a L1-contraction principle and we may apply the method of Gy\"ongy-Krylov to obtain stochastically strong solutions. These well-posedness results serve as a basis for the study of variational inequalities and Lewy-Stampacchia's inequalities for our problem.