We prove the existence of a volume preserving crystalline mean curvature flat flow starting from a compact convex set C ⊂ R and its convergence, modulo a time-dependent translation, to a Wulff shape with the corresponding volume. We also prove that if C satisfies an interior ball condition (the ball being the Wulff shape), then the evolving convex set satisfies a similar condition for some time...

We develop a well-posedness theory for solutions in L to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; a...

Let E ⊂ C be a compact set of positive logarithmic capacity. Let us suppose that for every polynomial P = id we have P −1 (E) = E. Then for all no constant polynomials f and g such that f −1 (E) = g −1 (E) we have f = g.