Pag 2 d i c h i a r a z i o n e s o s t i t u t i v a d i c e r t i f i c a z i o n e(art. 46 D.P.R. 445 del 28.12.2000)La/il sottoscritta/o……………………………………………………………………………………… nata/o a .

Abstract. Let the space R n be endowed with a Minkowski structure M (that is, M : R n → [0, +∞) is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class C 2 ), and let; y ∈ ∂Ω} be the Minkowski distance of a point x ∈ Ω from the boundary of Ω. We prove that a suitable extension of d Ω to R n (which plays the rôle of a signed Minkowski distance to ∂Ω) is of class C 2 in a tubular neighborhood of ∂Ω, and that d Ω is of class C 2 outside the cut locus of ∂Ω (that is, the closure of the set of points of nondifferentiability of d Ω in Ω). In addition, we prove that the cut locus of ∂Ω has Lebesgue measure zero, and that Ω can be decomposed, up to this set of vanishing measure, into geodesics starting from ∂Ω and going into Ω along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point x ∈ Ω outside the cut locus the pair (p(x), d Ω (x)), where p(x) denotes the (unique) projection of x on ∂Ω, and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

The system of partial differential equationsarises in the analysis of mathematical models for sandpile growth and in the context of the Monge-Kantorovich optimal mass transport theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the formwith f ≥ 0, and h ≥ 0 possibly non-convex, is also included.

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Abstract. We prove a uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.