The Neumann problem for a multivalued p-Laplace equation of Allen-Cahn type with a multiplicative stochastic force

Abstract

In this paper, we consider a parabolic problem with constraint written as a differential inclusion, driven by a multiplicative colored noise and involving a p-Laplace operator (for $p \geq 2$), nonlinear random source terms and subject to Neumann boundary conditions on a bounded Lipschitz domain of $R^d$ with $d \geq 1$. This contribution aims at proving existence and uniqueness of a solution for such a multivalued problem. On one hand, the existence result is proved by the analysis of a semi-implicit time discretization scheme constructed on a smoother version of our problem, itself obtained by a regularization "à la Moreau-Yosida" of the subdifferential term. The key point of our approach consists in finding a clever relation between the time step denoted $τ$ and the Moreau-Yosida regularization parameter denoted $ε$ in view to pass simultaneously to the limit with respect to $τ$ and $ε$. On the other hand, the uniqueness of the solution is proved by standard arguments.