$L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions

TL;DR

This paper establishes optimal $L^p$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, advancing regularity theory under minimal assumptions.

Contribution

It provides the first optimal global Calderón-Zygmund estimate for such equations with minimal regularity requirements.

Findings

Hessian of viscosity solutions is in $L^p$ space matching the nonhomogeneous term.

Achieves optimal regularity estimates for fully nonlinear parabolic equations with oblique boundary conditions.

Extends Calderón-Zygmund theory to a broader class of boundary value problems.

Abstract

We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calder\'{o}n-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in $L^{p}$ spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.