Lipschitz regularity in vectorial linear transmission problems

We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted the next phase. More exactly, given solution u:B1⊂Rn→Rm elliptic system div((A+(B−A)χD)∇u)=0inB1,where A B are Dini continuous, uniformly matrices, if ∇u∈L∞(D) then u Lipschitz in B1/2. similar result also derived for parabolic counterpart of this problem.