Lipschitz regularity in vectorial linear transmission problems. (August 2022) - Figalli, Alessio; Kim, Sunghan; Shahgholian, Henrik
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.