On new geometric conditions for some weakly lower semicontinuous functionals with applications to the rank-one conjecture of Morrey

We obtain new geometric necessary conditions for a function f to define a lower semicontinuous functional of the form If(u) = ∫Ωf(u)dx, where u satisfies a given conservation law, Pu = 0, defined by a differential operator P of degree one with constant coefficients. Those conditions imply the so-called Λ-convexity condition known as the rank-one condition when we deal with a functional of the calculus of variations. In particular, we derive some new geometric properties of quasi-convex functions and state some new questions related to the rank-one conjecture of Morrey.

On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities

In this paper we consider the following nonlinear parabolic variational inequality; u(t) ∈ D(Φ) for all where Δp is the so-called p-Laplace operator and Φ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data u0 and f.