Elliptic 1-Laplacian Equations with Dynamical Boundary Conditions

TL;DR

This paper studies an evolution problem involving the elliptic 1-Laplacian operator with dynamical boundary conditions, establishing existence, uniqueness, and comparison principles using nonlinear semigroup theory.

Contribution

It introduces a novel approach to analyze elliptic 1-Laplacian equations with dynamical boundary conditions, proving strong solutions and comparison results.

Findings

Existence and uniqueness of solutions established.

Solutions are shown to be strong solutions.

Comparison principles for solutions with different data proved.

Abstract

This paper is concerned with an evolution problem having an elliptic equation involving the 1-Laplacian operator and a dynamical boundary condition. We apply nonlinear semigroup theory to obtain existence and uniqueness results as well as a comparison principle. Our main theorem shows that the solution we found is actually a strong solution. We also compare solutions with different data.