Graziano Crasta

THE DISTANCE FUNCTION FROM THE BOUNDARY IN A MINKOWSKI SPACE

Let the space be endowed with a Minkowski structure (that is, is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain of class , let be the Minkowski distance of a point from the boundary of . We prove that a suitable extension of to (which plays the role of a signed Minkowski distance to ) is of class in a tubular neighborhood of , and that is of class outside the cut locus of (that is, the closure of the set of points of nondifferentiability of in ). In addition, we prove that the cut locus of has Lebesgue measure zero, and that can be decomposed, up to this set of vanishing measure, into geodesics starting from and going into 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 outside the cut locus the pair , where denotes the (unique) projection of on , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

Existence Results for Noncoerecive Variational Problems

The aim of this paper is to give an existence result for a class of one-dimensional, nonconvex, noncoercive problems in the calculus of variations. The main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity.

An existence result for the sandpile problem on flat tables with walls

We derive an existence result for solutions of a differential system which characterizes the equilibria of a particular model in granular matter theory, the so-called partially open table problem for growing sandpiles. Such result generalizes a recent theorem of [6] established for the totally open table problem. Here, due to the presence of walls at the boundary, the surface flow density at the equilibrium may result no more continuous nor bounded, and its explicit mathematical characterization is obtained by domain decomposition techniques. At the same time we show how these solutions can be numerically computed as stationary solutions of a dynamical two-layer model for growing sandpiles and we present the results of some simulations.

Kinetic Formulation and Uniqueness for Scalar Conservation Laws with Discontinuous Flux

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.

Structure of Solutions of Multidimensional Conservation Laws with Discontinuous Flux and Applications to Uniqueness

We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.

On the characterization of some classes of proximally smooth sets

We provide a complete characterization of closed sets with empty interior and positive reach in R 2 . As a consequence, we characterize open bounded domains in R 2 whose high ridge and cut locus agree, and hence C 1 planar domains whose normal distance to the cut locus is constant along the boundary. The latter result extends to convex domains in R n .

HYBRIDIZATION, NATURAL SELECTION, AND EVOLUTION OF REPRODUCTIVE ISOLATION: A 25-YEARS SURVEY OF AN ARTIFICIAL SYMPATRIC AREA BETWEEN TWO MOSQUITO SIBLING SPECIES OF THE Aedes mariae COMPLEX

Natural selection can act against maladaptive hybridization between co‐occurring divergent populations leading to evolution of reproductive isolation among them. A critical unanswered question about this process that provides a basis for the theory of speciation by reinforcement, is whether natural selection can cause hybridization rates to evolve to zero. Here, we investigated this issue in two sibling mosquitoes species, Aedes mariae and Aedes zammitii, that show postmating reproductive isolation (F1 males sterile) and partial premating isolation (different height of mating swarms) that could be reinforced by natural selection against hybridization. In 1986, we created an artificial sympatric area between the two species and sampled about 20,000 individuals over the following 25 years. Between 1986 and 2011, the composition of mating swarms and the hybridization rate between the two species were investigated across time in the sympatric area. Our results showed that A. mariae and A. zammitii have not completed reproductive isolation since their first contact in the artificial sympatric area. We have discussed the relative role of factors such as time of contact, gene flow, strength of natural selection, and biological mechanisms causing prezygotic isolation to explain the observed results.

Dynamics of mtDNA introgression during species range expansion: insights from an experimental longitudinal study

Introgressive hybridization represents one of the long-lasting debated genetic consequences of species range expansion. Mitochondrial DNA has been shown to heavily introgress between interbreeding animal species that meet in new sympatric areas and, often, asymmetric introgression from local to the colonizing populations has been observed. Disentangling among the evolutionary and ecological processes that might shape this pattern remains difficult, because they continuously act across time and space. In this context, long-term studies can be of paramount importance. Here, we investigated the dynamics of mitochondrial introgression between two mosquito species (Aedes mariae and Ae. zammitii ) during a colonization event that started in 1986 after a translocation experiment. By analyzing 1,659 individuals across 25 years, we showed that introgression occurred earlier and at a higher frequency in the introduced than in the local species, showing a pattern of asymmetric introgression. Throughout time, introgression increased slowly in the local species, becoming reciprocal at most sites. The rare opportunity to investigate the pattern of introgression across time during a range expansion along with the characteristics of our study-system allowed us to support a role of demographic dynamics in determining the observed introgression pattern.

A Chain Rule Formula in the Space BV and Applications to Conservation Laws

In this paper we prove a new chain rule formula for the distributional derivative of the composite function

A SYMMETRY PROBLEM FOR THE INFINITY LAPLACIAN

v(x)=B(x,u(x))