(A. Figalli), (S. Kim), (H. Shahgholian).l.https://doi.org/10.1016/j.na.2022.1129110362-546X/©2022 Published by Elsevier Ltd.

We consider vector-valued solutions to a linear transmission problem, and we provethat Lipschitz-regularity on one phase is transmitted to the next phase. Moreexactly, given a solutionu:B1 subset of Rn -> Rmto the elliptic systemdiv((A+ (B-A)chi D) backward difference u) = 0inB1,whereAandBare Dini continuous, uniformly elliptic matrices, we prove that if backward difference u is an element of L infinity(D)thenuis Lipschitz inB1/2. A similar result is also derived for theparabolic counterpart of this problem.(c) 2022 Published by Elsevier Ltd.

Automated summary

In this paper, we investigate vector-valued solutions to a linear transmission problem and prove the transmission of Lipschitz regularity from one phase to the next. We present the conditions for a solution to an elliptic system and demonstrate its Lipschitz property in a specific space. Similar results are also derived for the parabolic counterpart of the problem.