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

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} \;\; Ω, w &= 0 \;\;\;\text{on} \;\partial Ω, \end{aligned} \right. \end{equation*} \noindent where the domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ is convex, the matrix $(b_{ij}{)}_{n\times n}$ satisfies the uniform ellipticity conditions. For $g$ in the scaling critical Lorentz space $L(n,1)(\hat{I}\text{c◯})$, we establish boundary differentiability of solutions to the above problem. We also prove $C^{\mathrm{Log}-\mathrm{Lip}}$ regularity estimate at a boundary point in the case when $g\in L^n(\hat{I}\text{c◯})$.