Abstract. In this talk I will discuss the asymptotic analysis of a geometric functional introduced by Lauteri-Luckhaus for the analysis of grain boundaries in metals, i.e. the free boundaries between regions where a metal exhibits a (almost) perfect crystallization. The outcome of the analysis is the characterization of the line tension between grains with differently oriented crystalline structures, which agrees with the scaling found by Read and Shockly for small angle grain boundaries in polycrystals.

Abstract. We discuss some properties of minimizers of the W^s,1-fractional seminorm, which share similarities to their classical counterparts – functions of least gradient. Specifically, we examine the relationship between these minimizers and nonlocal minimal sets and use this connection to establish the existence of functions of least
W^s,1–seminorm. We further reason about the Euler-Lagrange equation which involves the fractional 1-Laplacian, and explore the existence of weak solutions by analyzing the asymptotics as p approaches 1 of the sequence of (s, p)-harmonic functions.

The results presented are obtained in collaboration with
S. Dipierro, L. Lombardini, J. Mazòn and E. Valdinoci.

Abstract. The talk concerns an ongoing work with L.De Luca and R.Scala about the asymptotic variational equivalence, at any logarithmic scaling regime, between Ginzburg-Landau energies (then Core Radius energies, as well) and suitable Mumford-Shah type energies. This extends a recent work by De Luca-Scala-Van Goethem which first employs such Mumford-Shah type functionals to approximate the energy of finitely many dislocations in a simplified topological setting.

Abstract. We will discuss two technical results on optimal transport penalized variational problems: the five gradients inequality and a maximum principle. The common trait is the idea to use competitors the dual problem in order to obtain non-trivial regularity properties for the primal one. These results have applications in particular in the JKO scheme, an heavily used time discretization for evolution problems.

Abstract. In this talk, I will present a recent result which establishes optimal regularity for isoperimetric sets with densities, under mild Holder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach the optimal regularity class C^1,α/(2-α) in any dimension.

This is a joint work with L. Beck and C. Seis.