## Lipschitz Regularity in Vectorial Linear Transmission Problems

created by figalli on 16 Mar 2021

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Submitted Paper

Inserted: 16 mar 2021
Last Updated: 16 mar 2021

Year: 2020

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 $${\rm div} ((A + (B-A)\chiD )\nabla u) = 0 \quad \text{in }B1,$$ 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.