The limit as $p\rightarrow\infty$ for the $p-$Laplacian equation with dynamical boundary conditions

TL;DR

This paper investigates the behavior of solutions to the $p$-Laplacian evolution problem with dynamical boundary conditions as $p$ approaches infinity, establishing convergence and linking it to optimal mass transportation.

Contribution

It proves Mosco convergence of the energy functional and characterizes the limit problem through optimal mass transportation, providing explicit solutions for specific cases.

Findings

Energy functional converges in the Mosco sense as p→∞.

Solutions to the evolution problem converge to a limit solution.

Limit problem relates to optimal mass transportation.

Abstract

In this paper we study the limit as $p\to \infty$ in the evolution problem driven by the $p-$Laplacian with dynamical boundary conditions. We prove that the natural energy functional associated with this problem converges to a limit in the sense of Mosco convergence and as a consequence we obtain convergence of the solutions to the evolution problems. For the limit problem we show an interpretation in terms of optimal mass transportation and provide examples of explicit solutions for some particular data.