A note on the density of the partial regularity result in the class of viscosity solutions

TL;DR

This paper demonstrates the density of partial regularity results within continuous viscosity solutions for fully nonlinear equations, enabling approximation of solutions under various conditions and extending to inhomogeneous problems with variable coefficients.

Contribution

It introduces a method to approximate viscosity solutions with partially regular solutions, broadening the applicability of regularity results in nonlinear PDEs.

Findings

Density of partial regularity results established in viscosity solutions.

Approximation sequences constructed for solutions under different operator conditions.

Applications demonstrated for inhomogeneous problems with variable coefficients.

Abstract

We establish the density of the partial regularity result in the class of continuous viscosity solutions. Given a fully nonlinear equation, we prove the existence of a sequence entitled to the partial regularity result, approximating its solutions. Distinct conditions on the operator driving the equation lead to density in different topologies. Our findings include applications to inhomogeneous problems, with variable-coefficients models.