An infinity Laplace equation with gradient term and mixed boundary conditions

TL;DR

This paper studies a modified infinity Laplace equation with a gradient term and mixed boundary conditions, establishing fundamental properties like existence, uniqueness, and stability through finite difference approximations.

Contribution

It introduces new results on the well-posedness of a modified infinity Laplace equation with gradient term and mixed boundary conditions, using comparison principles and finite difference methods.

Findings

Proved existence and uniqueness of solutions.

Established stability under boundary condition variations.

Validated the approach with finite difference approximations.

Abstract

We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation \[ -\Delta_\infty u - \beta |Du| = f, \] subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation.