cvgmt Seminarshttp://cvgmt.sns.it/seminars/en-usThu, 20 Sep 2018 01:12:35 +0000Well-posedness of ODE's in metric measure spaceshttp://cvgmt.sns.it/seminar/634/2018-09-24: <a href="/person/3/">L. Ambrosio</a>.<p>The DiPerna-Lions theory provides a good axiomatization to the notion of well-posedness for the ODE associated to a nonsmooth vector field, e.g. having a Sobolev but not Lipschitz regularity.</p><p>I will illustrate, with a reasonable number of details, my (relatively) recent work with D.Trevisan which extends this picture, so to speak, also to the case when even the ambient space is not so smooth. First I will revisit the Di Perna-Lions theory, then I will explain how in this broader context the notions ``Sobolev vector field'' and ''solution to the ODE'' have to be properly understood. For the first one we achieve the goal using Gamma-calculus tools (and so the natural context will be Dirichlet forms, Markov semigroups, Carr\'e du Champ operators) for the second one we use ideas coming from Optimal Transport and Geometric Measure Theory, the so-called superposition principle.</p>http://cvgmt.sns.it/seminar/634/The isoperimetric problem in the Euclidean space with densitieshttp://cvgmt.sns.it/seminar/635/2018-09-24: <a href="/person/151/">A. Pratelli</a>.<p>In this short course we will present the isoperimetric problem in a Euclidean space with density. In other words, one wants to minimize, as usual, the perimeter of sets with a given volume; however, the "perimeter" and the "volume" of a set are given by the integral of two given functions (the densities) on the set and on its boundary respectively. The classical case clearly corresponds to the choice of both functions constantly equal to $1$.</p><p>This is a quite classical question, but there are also very recent developments. We will try to give a general overview of the subject, and to explain what is known and some of the most important open problems which are left. We will also describe some of the key ideas of the proofs, without entering too much into the technical details.</p><p>In the first of the three lessons we will discuss the problem, give an idea of the classical known results, and individuate the main questions that one wants to answer. Then, roughly speaking, the second lesson will be devoted to study the questions concerning the regularity of isoperimetric sets, and the last one the questions concerning the existence.</p>http://cvgmt.sns.it/seminar/635/How a minimal surface leaves a thin obstaclehttp://cvgmt.sns.it/seminar/636/2018-09-24: E. Spadaro.<p>Free boundary problems naturally arise in several applications in mathematical physics, biology and engineering. They are characterized by different sets of differential relations on distinct domains which are not assigned apriori but are among the unknowns of the problem. </p><p>This course will deal with the non-parametric thin obstacle problem for minimal surfaces, which is a prototypical example of a free boundary problem with higher co-dimension free boundary. This problem consists in minimizing the area of a graph with prescribed boundary conditions and with a unilateral constrain imposed on a hyperplane. </p><p>In the lectures I will present some of the key-ideas that have been introduced to treat this and similar problems, with a special focus (time permitting) on the new variational and measure theoretic techniques recently developed.</p>http://cvgmt.sns.it/seminar/636/Asymptotic stability of the gradient flow of nonlinear energieshttp://cvgmt.sns.it/seminar/655/2018-09-25: <a href="/person/217/">N. Fusco</a>.<p>We shall discuss short-time existence of the surface diffusion equation in 3D with a nonlocal elastic term. We will also show that strictly stable stationary sets are asymptotically stable.</p>http://cvgmt.sns.it/seminar/655/Geometry of sets of non-differentiabilityhttp://cvgmt.sns.it/seminar/656/2018-09-26: O. Maleva.<p>There are many subsets $N$ of $\mathbb{R}^n$ for which one can find a real-valued Lipschitz function $f$ defined on the whole of $\mathbb{R}^n$ but non-differentiable at every point of $N$. Of course, by the Rademacher theorem any such set $N$ is Lebesgue null, however, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of $\mathbb{R}^n$ gives rise to such a Lipschitz function $f$. In this talk, I explain that a sufficient condition on a set $N$ for such $f$ to exist is to be locally unrectifiable with respect to curves in a cone of directions. In particular, every $1$-purely unrectifiable set $U$ possesses a Lipschitz function non-differentiable on $U$ in the strongest possible sense. I also give an example of a universal differentiability set unrectifiable with respect to a fixed cone of directions, showing that one cannot relax the conditions.This is joint work with D. Preiss.</p>http://cvgmt.sns.it/seminar/656/Mass-minimizing integral currents: regularity at the boundaryhttp://cvgmt.sns.it/seminar/657/2018-09-26: <a href="/person/866/">A. Massaccesi</a>.<p>In this seminar I will review (a part of) the history of the Plateau problem and the main regularity theorems for mass-minimizing currents. In the last part of the seminar I will explain the content of a forthcoming paper by De Lellis, De Philippis, Hirsch and myself establishing generic regularity for the boundary points of a mass-minimizing current.</p>http://cvgmt.sns.it/seminar/657/Generic regularity of free boundaries for the obstacle problem in $\mathbb{R}^3$http://cvgmt.sns.it/seminar/658/2018-09-26: <a href="/person/1526/">X. Ros-Oton</a>.<p>Free boundary problems are those described by PDE's that exhibit a priori unknown (free) interfaces or boundaries. The obstacle problem is the most classical and motivating example in the study of free boundary problems. A milestone in this context is the classical work of Caffarelli (Acta Math. 1977), in which he established for the first time the regularity of free boundaries in the obstacle problem, outside a certain set of singular points. A long-standing open question in the field asks to establish generic regularity results in this setting (e.g. to prove that for almost every boundary data there are no singular points). This type of questions arise as well in many other nonlinear PDE's and in Geometric Analysis. The goal of this talk is to present some new results in this context, proving in particular the generic regularity of free boundaries for the obstacle problem in $\mathbb{R}^3$. This is a joint work with A. Figalli and J. Serra.</p>http://cvgmt.sns.it/seminar/658/Rectifiability issues in sub-Riemannian geometryhttp://cvgmt.sns.it/seminar/660/2018-09-26: <a href="/person/256/">D. Vittone</a>.<p>In this talk we discuss two problems concerning "rectifiability" in sub-Riemannian geometry and particularly in the model setting of Carnot groups. The first problem regards the rectifiability of boundaries of sets with finite perimeter in Carnot groups, while the second one concerns Rademacher-type results (existence of a tangent plane out of a negligible set) for (intrinsic) graphs with (intrinsic) Lipschitz regularity. We will introduce both problems and discuss the state-of-the-art. Eventually, we will present some recent results about the rectifiability of sets with finite perimeter in a certain class of Carnot groups (including the simplest open case, i.e., the Engel group) and about a Rademacher theorem for intrinsic Lipschitz graphs in Heisenberg groups.</p>http://cvgmt.sns.it/seminar/660/Analysis of Novel Domain Wall Types in Ferromagnetic Nanostructureshttp://cvgmt.sns.it/seminar/651/2018-09-26: C. Muratov.<p>Recent advances in nanofabrication allow an unprecedented degree of control of ferromagnetic materials down to the atomic scale, resulting in novel nanostructures whose properties are often dominated by material interfaces. Mathematically, these systems give rise to challenging problems in the calculus of variations that feature non-convex, vectorial, topologically constrained, multi-scale variational problems. Yet despite the daunting complexity inherent in the problem arising from the 21st century technological applications, rigorous variational analysis can still elucidate energy-driven pattern formation in these systems. In this talk, I will discuss several examples of variational problems emerging from models of current ferromagnetic nanostructures under development. With the help of asymptotic techniques and explicit solutions, I will give three examples in which the energy minimizing configurations may be characterized in terms of optimal one-dimensional transition layer profiles separating magnetic domains with different magnetization orientation.</p>http://cvgmt.sns.it/seminar/651/