Lipschitz regularity in vectorial linear transmission problems

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→Rm to the elliptic system div((A+(B−A)χD)∇u)=0inB1,where A and B are Dini continuous, uniformly elliptic matrices, we prove that if ∇u∈L∞(D) then u is Lipschitz in B1/2. A similar result is also derived for the parabolic counterpart of this problem.