A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

TL;DR

This paper presents a straightforward proof of a strong comparison principle for semicontinuous viscosity solutions to the prescribed mean curvature equation, overcoming challenges posed by its non-uniform ellipticity and dependence on spatial variables.

Contribution

The paper introduces a simple proof of the strong comparison principle using only basic concepts like viscosity solutions, convolutions, and classical elliptic theory, applicable to a complex nonlinear PDE.

Findings

Established a strong comparison principle for semicontinuous viscosity solutions.

Proved a weak comparison principle in bounded domains.

Simplified the proof process using fundamental PDE tools.

Abstract

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.