Lipschitz regularity in vectorial linear transmission problems. (August 2022)

• Record Type: Journal Article
• Authors: Figalli, Alessio; Kim, Sunghan; Shahgholian, Henrik
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: primary 35B65 35J47 35K40 -- secondary 35R35; Transmission problems -- Elliptic systems -- Parabolic systems -- Lipschitz regularity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112911
• Languages: English
• ISSNs: 0362-546X

Abstract

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 ⊂ R n → R m to the elliptic system div ( ( A + ( B − A ) χ D ) ∇ u ) = 0 in B 1, 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.