On the comparison principle for second order elliptic equations without first and zeroth order terms

TL;DR

This paper introduces a unifying structural condition for the comparison principle in second order elliptic equations, simplifying proofs and extending existing theories in viscosity solutions.

Contribution

It presents a new structural condition on the operator that unifies and simplifies the comparison principle theory for elliptic equations without first and zeroth order terms.

Findings

A new unifying structural condition for the comparison principle.

Simplified proofs of classical comparison results.

Extension of the comparison principle to broader classes of equations.

Abstract

We consider the comparison principle for semicontinuous viscosity sub- and supersolutions of second order elliptic equations on the form $F(D^2 w,x) = 0$. A structural condition on the operator is presented that seems to unify the different existing theories. A new result is obtained and the proofs of the classical results are simplified.