Lipschitz Regularity in Vectorial Linear Transmission Problems

2020/12/31 by Alessio Figalli, Sunghan Kim, Figalli, Alessio +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2012.15499

openalex publication_date 2020/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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:B₁⊂ ℝⁿ → ℝ^m to the elliptic system div ((A + (B-A)χ_D )∇ u) = 0 in B₁, where A and B are Dini continuous, uniformly elliptic matrices, we prove that if ∇ u ∈ L^∞ (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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