Explicit Singular Viscosity Solutions of the Aronsson Equation

TL;DR

This paper constructs explicit viscosity solutions to the Aronsson equation in higher dimensions, revealing the necessity of strict level convexity for C^1 regularity and characterizing the solutions' structure.

Contribution

It provides explicit solutions when level sets contain line segments and clarifies the regularity conditions needed for solutions to be C^1.

Findings

Explicit solutions exist under certain conditions

Strict level convexity is necessary for C^1 regularity

Solutions are superpositions of linear and singular parts

Abstract

We establish that when n >= 2 and H is a C^1 Hamiltonian such that some level set contains a line segment, the Aronsson equation admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz continuous everywhere differentiable singular part which in general is non-C^1 and nowhere twice differentiable. In particular, we supplement the C^1 regularity result of Wang and Yu by deducing that strict level convexity is necessary for C^1 regularity of solutions.