Lipschitz Regularity in Vectorial Linear Transmission Problems

TL;DR

This paper proves that Lipschitz regularity of solutions in one phase of a vectorial linear transmission problem extends to the entire domain, including the interface, for both elliptic and parabolic cases.

Contribution

It establishes Lipschitz regularity transmission across phases for vector-valued solutions in linear transmission problems with Dini continuous coefficients.

Findings

Lipschitz regularity in one phase implies Lipschitz regularity in the whole domain.

Results apply to both elliptic and parabolic transmission problems.

Regularity transfer holds under Dini continuity and ellipticity conditions.

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)\chi_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.