A $C^1$ regularity result for the inhomogeneous normalized infinity Laplacian

TL;DR

This paper proves that solutions to a specific inhomogeneous infinity Laplacian problem on convex domains are continuously differentiable, using power-concavity as an intermediate step.

Contribution

It establishes a $C^1$ regularity result for solutions of the inhomogeneous normalized infinity Laplacian, a novel regularity insight for this class of PDEs.

Findings

Solution is of class $C^1$ on convex domains.

Power-concavity of the solution with exponent 1/2 is demonstrated.

Unique solution exists for the Dirichlet problem with constant source.

Abstract

We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.