Convergence of solutions to the $p$-Laplace evolution equation as $p$ goes to 1

TL;DR

This paper proves that solutions to the p-Laplace evolution equation continuously depend on p and initial data, including the highly singular case p=1, with solutions converging strongly in L^2 as p approaches 1.

Contribution

It establishes the continuous dependence and convergence of solutions to the p-Laplace evolution equation as p approaches 1, encompassing the singular limit case.

Findings

Solutions depend continuously on p and initial data.

Strong convergence of solutions in L^2 as p approaches 1.

Includes the singular case p=1 in the analysis.

Abstract

We prove that the set of solutions to the parabolic singular $p$-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain $\Omega$ for all space dimensions is continuous in the parameter $p\in [1,+\infty)$ and the initial data. The highly singular limit case p=1 is included. In particular, we show that the solutions $u_p$ converge strongly in $L^2(\Omega)$, uniformly in time, to the solution $u_1$ of the parabolic 1-Laplace equation as $p\to 1$.