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:

$$\begin{aligned} \left\{ \begin{aligned} - b_{ij} \, \frac{\partial ^{2} w}{\partial x_{i} \partial x_{j}}&= g \,\,\, \text {in} \,\, \Omega , \\ w&= 0 \,\,\,\text {on} \,\partial \Omega , \end{aligned} \right\} . \end{aligned}$$

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