Superposition Property in Disjoint Variables for the Infinity Laplace Equation

TL;DR

This paper proves a superposition principle for the inhomogeneous infinity-Laplace equation, showing that solutions in separate domains can be combined to form a solution in the product domain, aiding regularity analysis.

Contribution

It establishes a general superposition property for viscosity solutions of the infinity-Laplace equation in disjoint variables, extending previous results and exploring metric space generalizations.

Findings

Sum of solutions in separate domains is a solution in the product domain

Superposition principle simplifies regularity analysis for inhomogeneous equations

Extension of results to metric spaces using cone comparison techniques

Abstract

We establish a superposition principle in disjoint variables for the inhomogeneous infinity-Laplace equation. We show that the sum of viscosity solutions of the inhomogeneous infinity-Laplace equation in separate domains is a viscosity solution in the product domain. This result has been used in the literature with certain particular choices of solutions to simplify regularity analysis for a general inhomogeneous infinity-Laplace equation by reducing it to the case without sign-changing inhomogeneous terms and vanishing gradient singularities. We present a proof of this superposition principle for general viscosity solutions. We also explore generalization in metric spaces using cone comparison techniques and study related properties for general elliptic and convex equations.