Boundary differentiability of solutions to elliptic equations in convex domains in the borderline case

TL;DR

This paper investigates the boundary differentiability of solutions to elliptic equations in convex domains under critical conditions, establishing new regularity results in borderline function spaces.

Contribution

It proves boundary differentiability for solutions with data in the critical Lorentz space and introduces $C^{ ext{Log-Lip}}$ regularity estimates at boundary points.

Findings

Boundary differentiability established for data in $L(n,1)$.

$C^{ ext{Log-Lip}}$ regularity at boundary points for $g \

Solutions are differentiable at the boundary under critical conditions.

Abstract

In this work, we consider the following elliptic partial differential equations: \begin{equation*} \left\{ \begin{aligned}{} - b_{ij} \; \frac{\partial^{2} w}{\partial x_{i} \partial x_{j}} &= g \;\;\; \text{in} \;\; \Omega, w &= 0 \;\;\;\text{on} \;\partial \Omega, \end{aligned} \right. \end{equation*} \noindent where the domain $\Omega \subset \mathbb{R}^{n}$ is convex, the matrix $\big(b_{ij}\big)_{n \times n}$ satisfies the uniform ellipticity conditions. For $g$ in the scaling critical Lorentz space $ L(n,\; 1)(\Omega)$, we establish boundary differentiability of solutions to the above problem. We also prove $C^{\mathrm{Log-Lip}}$ regularity estimate at a boundary point in the case when $g \in L^{n}(\Omega)$.