Global weighted Lorentz estimates of oblique tangential derivative problems for weakly convex fully nonlinear operators

TL;DR

This paper establishes weighted Lorentz-Sobolev estimates for solutions of fully nonlinear elliptic equations with oblique boundary conditions, extending the theory under weakened convexity assumptions and applying to obstacle problems.

Contribution

It introduces new weighted Lorentz estimates for viscosity solutions of fully nonlinear elliptic equations with oblique boundary conditions under relaxed convexity conditions.

Findings

Derived Lorentz-Sobolev estimates for solutions with oblique boundary conditions

Extended estimates to obstacle problems and related applications

Provided conditions on data for the estimates to hold

Abstract

In this work, we develop weighted Lorentz-Sobolev estimates for viscosity solutions of fully nonlinear elliptic equations with oblique boundary condition under weakened convexity conditions in the following configuration $F(D^{2}u, Du, u, x) = f(x)$ in $\Omega$ and $\beta\cdot Du+\gamma u=g$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ ($n \geq 2$), under suitable assumptions on the source term f, data $\beta$, $\gamma$ and g. In addition, we obtain Lorentz-Sobolev estimates for solutions to the obstacle problem and others applications.