Inhomogeneous Dirichlet problems involving the infinity-Laplacian

TL;DR

This paper studies the existence of viscosity solutions for inhomogeneous Dirichlet problems involving the infinity-Laplacian, providing new sufficient conditions and examples of non-existence in certain cases.

Contribution

It offers a comprehensive account of existence conditions for solutions to the inhomogeneous infinity-Laplacian Dirichlet problem, improving upon previous results and identifying cases with no solutions.

Findings

Sufficient conditions for existence of solutions are established.

Examples show these conditions cannot be relaxed.

Certain inhomogeneous terms lead to no solutions in large domains.

Abstract

Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $\Delta_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of Perron's method and a priori estimates to give general sufficient conditions on the right hand side $f$ that would ensure existence of viscosity solutions to the Dirichlet problem. Examples show that these sufficient conditions may not be relaxed. We also identify a class of inhomogeneous terms for which the corresponding Dirichlet problem has no solution in any domain with large in-radius. Several results, which are of independent interest, are developed to build towards the main results. The existence theorems provide substantial improvement of previous results, including our earlier results \cite{BMO} on this topic.