Evolution problem for the $1$-Laplacian with mixed boundary conditions

TL;DR

This paper studies the evolution problem for the 1-Laplacian with mixed boundary conditions, proving existence and uniqueness of strong solutions using maximal monotone operator theory, and relates boundary flux to large solutions.

Contribution

It establishes existence and uniqueness of strong solutions for the 1-Laplacian evolution problem with mixed boundary conditions, including boundary flux characterization and explicit examples.

Findings

Existence and uniqueness of strong solutions for data in L^2(Ω)

Boundary flux of 1 leads to large solutions

Explicit examples of solutions provided

Abstract

This paper deals with evolution problem for the $1$-Laplacian with mixed boundary conditions on a bounded open set $\Omega$ of $\R^N$. We prove existence and uniqueness of strong solutions for data in $L^2(\Omega)$ by mean of the theory of maximal monotone operator. We also see that if the flux on the boundary is~$1$ (that is, the maximum possible) then these strong solutions can be seen as the large solutions introduced in \cite{MP}. We give explicit examples of solutions.