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} \;\; \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 $g \in L^{n}(\Omega)$.