Calculus of Variations & Geometric Measure Theory

Contribution List

31. Free boundary regularity for the obstacle problem

Alessio Figalli (ETH Zürich)

12/06/2023, 10:00

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to...

39. Evolution of probability measures: beyond gradient flows

Giuseppe Savaré (Università Bocconi, Milano)

12/06/2023, 11:30

The theory of contraction semigroups in the Wasserstein space of Euclidean probability measures generated by displacement convex functionals is well understood: the variational approach relies on convergence estimates on the Jordan-Kinderlehrer-Otto/Minimizing Movement scheme and on the non-smooth infinitesimal Riemannian structure of the space. In particular, the contraction property of the...

29. Beyond the classical Cauchy-Born rule

Andrea Braides (SISSA, Trieste)

12/06/2023, 14:30

Physically motivated variational problems involving non-convex energies are often formulated in a discrete setting and contain boundary conditions. The long-range interactions in such problems, combined with constraints imposed by lattice discreteness, can give rise to the phenomenon of geometric frustration even in a one-dimensional setting. While non-convexity entails the formation of...

46. Wassertein Sobolev spaces: numerics and deep learning over P2

Massimo Fornasier (TUM München)

12/06/2023, 16:00

We start the talk by presenting general results of strong density of sub-algebras of bounded
Lipschitz functions in metric Sobolev spaces. We apply such results to show the density of smooth cylinder functions in Sobolev spaces of functions on the Wasserstein space $\mathcal P_2$ endowed with a finite positive Borel measure. As a byproduct, we obtain the infinitesimal Hilbertianity of...

25. Min-Max construction of anisotropic minimal surfaces

Guido De Philippis (New York University)

13/06/2023, 09:00

Since the seminal work of Birkhoff min max techniques have been a powerful tool in showing existence of stationary point of geometric variational problem. Almgren-Pitts developed in the 80`s extended the min-max framework to the area functional, to show existence of minimal surfaces. In this talk I will discuss the extension of the Almgren-Pitts theory to anisotropic energies. The lack of...

40. Local and global minimizers for a capillarity type problem

Nicola Fusco (Università degli Studi di Napoli Federico II)

13/06/2023, 10:30

I will present a model for vapor-liquid-solid growth of nanowires where liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. I will first discuss global existence of minimizers and then, in the case of rotationally symmetric nanowires, I will explain how the presence of a sharp edge...

28. Variational approach for grain boundaries

Adriana Garroni (Università La Sapienza Roma)

13/06/2023, 11:30

The talk will focus on the role played by material defects in determination of the surface tension carried by grain boundaries in polycrystals. Defects induce elastic distortion in the bulk, therefore a variational model accounting for both, elastic energy and presence of defects, as the one proposed by Lauteri and Luckhaus, is the starting point for our analysis.

37. The Riemannian Penrose inequality via nonlinear potential theory

Carlo Mantegazza (SNS Pisa)

13/06/2023, 14:30

I will discuss the Riemannian Penrose inequality in an asymptotically flat 3-manifold with nonnegative scalar curvature, and the main points of a new proof by means of a monotonicity formula, holding along the level sets of the p-capacitary potentials of the horizon of a black hole.

Joint work with Francesca Oronzio.

41. Frobenius theorem for nonsmooth surfaces

Giovanni Alberti (Università di Pisa)

13/06/2023, 16:00

A classical result in geometry, Frobenius theorem, states that there exist no $k$-dimensional surface which is tangent to a non-involutive distribution of $k$-planes $V$.
One may wonder to which extent this statement can be generalized to weaker notions of surfaces, such as rectifiable sets and currents.

Following the work of Z. Balogh and S. Delladio, an interesting class is that of...

45. Stable solutions to semilinear elliptic equations

Xavier Cabré (Universidad Politécnica de Cataluña)

14/06/2023, 09:00

The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. It was initiated by a work of Crandall and Rabinowitz, motivated by the Gelfand problem in combustion theory. The theory experienced a revival in the mid-nineties after new progress made by Brezis and collaborators. I will present these developments, as well as a recent work, in collaboration with...

43. Mean value formulas for degenerate parabolic equations

Diego Pallara (Università del Salento)

14/06/2023, 10:30

I deal with mean value formulas for classical solutions of some degenerate parabolic equation with $C^1$ coefficients, including degenerate Kolmogorov equations. The theory of functions of bounded variation and perimeters in stratified Lie groups is an essential tool in the proof.

34. Timelike Ricci bounds and Einstein's theory of gravity in a non smooth setting: an optimal transport approach

Andrea Mondino (University of Oxford)

14/06/2023, 11:30

Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been as a non-smooth extension of Riemannian manifolds). Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in...

44. Sobolev regularity of flows associated to vector fields with exponential or sub-exponential summability

Francesco Serra Cassano (Università di Trento)

14/06/2023, 14:30

We are concerned with the Sobolev regularity of a flow $X:\,I\times I\times \Omega\to\Omega$ associated to a non-smooth vector field $b:\,I\times\Omega\to\Omega$, i.e. the solution of the Cauchy problem
$$ \begin{cases} \partial_tX(t,s,x)&=\,b(t,X(t,s,x))\\ X(s,x)&=\,x \end{cases} \quad\,t,s\in I, \,x\in\Omega\,, \tag{P} $$ where $\Omega\subset\mathbb R^n$ is a given open domain and...

33. Numerical approximation of mean curvature flows using neural networks

Simon Masnou (UCBL Lyon)

14/06/2023, 16:00

The talk will focus on new neural network-based numerical methods for approximating the mean curvature flow of general interfaces, both oriented and non-oriented. To learn the correct evolution law, the networks are trained on implicit representations of exact interface evolutions. They have a very simple structure inspired by some splitting schemes used for the discretization of the...

24. Phase Separation in Heterogeneous Media

Irene Fonseca (Carnegie Mellon University)

15/06/2023, 09:00

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may have to depend on the...

36. Hyperbolic nonsmooth calculus

Nicola Gigli (SISSA Trieste)

15/06/2023, 10:30

In the last 25 years, tremendous progresses have been made in the field of non-smooth analysis: after the pioneering works of the Finnish school and Cheeger’s seminal contribution, interest on the topic has been revamped by Lott-Sturm-Villani’s papers on weak lower Ricci curvature bounds.

More recently, there has been a surging interest in non-smooth "hyperbolic" geometry, i.e. in spaces...

42. Antagonistic cost functionals in shape optimization

Giuseppe Buttazzo (Università di Pisa)

15/06/2023, 11:30

In several shape optimization problems one has to deal with cost functionals of the form
${\mathcal F}(\Omega)=F(\Omega)+kG(\Omega)$, where $F$ and $G$ are two shape functionals with a different monotonicity behavior and $\Omega$ varies in the class of domains with prescribed measure. In particular, the cost functional ${\mathcal F}(\Omega)$ is not monotone with respect to $\Omega$ and the...

47. Concave Random Assignment Problem in One Dimension: Kantorovich Meets Young

Dario Trevisan (Università di Pisa)

15/06/2023, 14:30

We investigate the one-dimensional random assignment problem in the
concave case, where the assignment cost is determined by a concave power
function of the distance between n source and n target points that are
i.i.d.\ random variables. Unlike the convex case, the optimal assignment
in the concave case can depend on the power parameter and exhibits a
multi-scale structure. We settle some...

38. Singular Weyl's law with Ricci curvature bounded below

Shouhei Honda (Tohoku University)

15/06/2023, 15:50

We discuss the asymptotic behavior of eigenvalues of the Laplacian on a compact finite dimensional metric measure space with Ricci curvature bounded below, so-called an RCD space. Ambrosio and Tewodrose with me gave a necessary and sufficient condition for the validity of the standard Weyl's law in terms of the size of the regular set. The main purpose of this talk is to provide examples of...

35. Homogenisation of free discontinuity problems

Gianni Dal Maso (SISSA Trieste)

16/06/2023, 09:00

We study deterministic and stochastic homogenisation problems for free discontinuity functionals under hypotheses which lead to an interaction between surface and volume energies. The results are based on a compactness theorem with respect to Gamma-convergence, on the characterisation of the integrands of the Gamma-limit by means of limits of minimum values of some auxiliary minimum problems...

30. Singularities of solutions to Hamilton-Jacobi equations: from PDE’s to topology, passing through geometric measure theory

Piermarco Cannarsa (Università degli Studi di Roma Tor Vergata)

16/06/2023, 10:30

The study of the structural properties of the set of points at which a solution u of a first order Hamilton-Jacobi equation fails to be differentiable -in short, the singular set of u- has been the subject of a long-term project that started in the late sixties with a seminal paper by W. H. Fleming. Research on such a topic picked up again after the introduction of viscosity solutions by...

26. On the apparent non-convexity of several generalizations of optimal transport

Yann Brenier (École normale supérieure Paris)

16/06/2023, 11:30

The optimal transport problem benefits from several convex formulations which make it well suited to mathematical analysis. Nevertheless, several interesting generalizations (with applications in particular to Einstein's and Schroedinger's equations) involving complex-valued vector or matrix density fields apparently lose this convex character which makes their analysis much more problematic....