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

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 proces...

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Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Sapountzoglou, Niklas, Tahraoui, Yassine, Vallet, Guy, Zimmermann, Aleksandra
Format: Article
Language:English
Subjects:
Barriers
Boundary conditions
Boundary value problems
Convergence
Dirichlet problem
Existence theorems
Inequalities
Martingales
Nonlinearity
Singular perturbation
Well posed problems
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Summary: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. Therefore 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.
ISSN:2331-8422