On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions

TL;DR

This paper demonstrates that for fully nonlinear parabolic equations with measurable coefficients, one can construct an approximating equation that admits a unique, well-behaved solution, even without convexity assumptions.

Contribution

It introduces a method to approximate fully nonlinear parabolic equations with solutions that have bounded derivatives, bypassing the need for convexity assumptions.

Findings

Existence of a unique continuous solution for the approximating equation.

Bounded first derivatives and locally bounded second derivatives of solutions.

Approximation modifies the original equation only for large values of the solution and its derivatives.

Abstract

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.