Published Paper

Inserted: 16 mar 2021
Last Updated: 19 aug 2024

Journal: Nonlinear Anal.
Year: 2022

Abstract:

We consider vector-valued solutions to a linear transmission problem and prove that Lipschitz regularity on one phase is transmitted to the next phase. More precisely, given a solution \( u : B_1 \subset \mathbb{R}^n \to \mathbb{R}^m \) to the elliptic system \[ \text{div} \left( (A + (B - A) \chi_D) \nabla u \right) = 0 \quad \text{in } B_1, \] 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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