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Well-posedness of a time discretization scheme for a stochastic p-Laplace equation with Neumann boundary conditions

Caroline Bauzet, Kerstin Schmitz, Cédric Sultan, Aleksandra Zimmermann

TL;DR

The paper addresses the well-posedness of a semi-implicit Euler–Maruyama time discretization for a parabolic SPDE with a $p$-Laplacian under Neumann boundary conditions and multiplicative colored noise. It regularizes the maximal monotone constraint by a Moreau–Yosida approximation and proves, via the Minty–Browder framework, that each time-step equation admits a unique solution in the variational space $V=W^{1,p}(D)$ under a time-step restriction $\tau<1/L_{\beta}$. The analysis combines coercivity, strict monotonicity, hemicontinuity of the discrete operator $A_{\varepsilon,\tau}$, and Pettis measurability to ensure both existence and $\mathcal{F}_{t_{n+1}}$-measurability of the iterates. This establishes a solid foundation for future convergence studies towards the stochastic p-Laplace–Allen–Cahn type problem with the maximal monotone constraint, with potential applications in solid mechanics.

Abstract

In this contribution, we are interested in the analysis of a semi-implicit time discretization scheme for the approximation of a parabolic equation driven by multiplicative colored noise involving a $p$-Laplace operator (with $p\geq 2$), nonlinear source terms and subject to Neumann boundary conditions. Using the Minty-Browder theorem, we are able to prove the well-posedness of such a scheme.

Well-posedness of a time discretization scheme for a stochastic p-Laplace equation with Neumann boundary conditions

TL;DR

The paper addresses the well-posedness of a semi-implicit Euler–Maruyama time discretization for a parabolic SPDE with a -Laplacian under Neumann boundary conditions and multiplicative colored noise. It regularizes the maximal monotone constraint by a Moreau–Yosida approximation and proves, via the Minty–Browder framework, that each time-step equation admits a unique solution in the variational space under a time-step restriction . The analysis combines coercivity, strict monotonicity, hemicontinuity of the discrete operator , and Pettis measurability to ensure both existence and -measurability of the iterates. This establishes a solid foundation for future convergence studies towards the stochastic p-Laplace–Allen–Cahn type problem with the maximal monotone constraint, with potential applications in solid mechanics.

Abstract

In this contribution, we are interested in the analysis of a semi-implicit time discretization scheme for the approximation of a parabolic equation driven by multiplicative colored noise involving a -Laplace operator (with ), nonlinear source terms and subject to Neumann boundary conditions. Using the Minty-Browder theorem, we are able to prove the well-posedness of such a scheme.
Paper Structure (6 sections, 2 theorems, 21 equations)

This paper contains 6 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem and assumptions
  3. State of the art and aim of the study
  4. Semi-implicit time discretization scheme
  5. Presentation of the scheme and main result
  6. Proof of Proposition \ref{['well-posedness_scheme']}

Key Result

Proposition 1

Let us assume that Hypotheses $(H_1)$ to $(H_5)$ are satisfied, consider a fixed parameter $\varepsilon>0$, a fixed $M\in \mathbb{N}^\ast$, define $\tau=T/M$, $t_n=n\tau$ ($\forall n\in\llbracket 0,M-1\rrbracket$) and $u_{\epsilon,0}=u_0$. Then, for any given $\mathcal{F}_{t_n}$-measurable random va

Theorems & Definitions (4)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Lemma 2: Algebraic inequality