Lipschitz Regularity in Vectorial Linear Transmission Problems

Abstract

We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution $u:B_1\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ to the elliptic system \begin{equation*} \mbox{div} ((A + (B-A)χ_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous, uniformly elliptic matrices, we prove that if $\nabla u\in L^{\infty }(D)$ then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.

25 pages