On boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations

TL;DR

This paper proves boundary H"older gradient estimates for solutions to linearized Monge-Amp ext{`}ere equations with less regular data, extending previous regularity results to broader conditions.

Contribution

It establishes new boundary gradient estimates for solutions with $L^{p}$ right hand side and $C^{1, ext{ extgamma}}$ boundary data, broadening the scope of regularity theory.

Findings

Boundary H"older gradient estimates are valid under $L^{p}$ right hand side.

Results extend previous regularity to less smooth boundary data.

Estimates depend on natural assumptions on domain and boundary conditions.

Abstract

In this paper, we establish boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,\gamma}$ boundary values under natural assumptions on the domain, boundary data and the Monge-Amp\`ere measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Amp\`ere equations with bounded right hand side and $C^{1, 1}$ boundary data.