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.

We show that in two dimensions or higher, the Mordukhovich-loffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with ...

Given a lower semicontinuous function f : Rn → R ∪ {+∞}, we prove that the set of points of Rn where the lower Dini subdifferential has convex dimension k is countably (n − k)-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.