Published Paper

Inserted: 16 mar 2021
Last Updated: 10 feb 2023

Journal: Nonlinear Anal.
Year: 2022

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 $$ {\rm div} ((A + (B-A)\chi_D )\nabla u) = 0 \quad \text{in }B_1, $$ 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.

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• Lipschitz-regularity-in-vectorial-linear-transmission-problems.pdf