An ersatz existence theorem for fully nonlinear parabolic equations without convexity assumptions

TL;DR

This paper constructs an approximation for fully nonlinear parabolic equations without convexity assumptions, ensuring solutions with controlled derivatives, thus extending existence results under minimal regularity conditions.

Contribution

It introduces a novel approximation method that modifies the original equation only for large second derivatives, bypassing the need for convexity assumptions.

Findings

Existence of continuous solutions with bounded derivatives in whole space

Approximate equations differ only for large second derivatives

Method applies to equations with minimal regularity assumptions

Abstract

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in the whole space or in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a continuous solution with the first and the second spatial derivatives under control: bounded in the case of the whole space and locally bounded in case of equations in cylinders. The approximating equation is constructed in such a way that it modifies the original one only for large values of the second spatial derivatives of the unknown function. This is different from a previous work of Hongjie Dong and the author where the modification was done for large values of the unknown function and its spatial derivatives.