Abstracts Spring Semester 2024

Elena Beretta (NYU Abu Dhabi)

Title: Identification of cavities in a nonlinear model arising from cardiac electrophysiology

Abstract: Detecting ischemic regions is crucial for preventing lethal ventricular ischemic tachycardia. This is typically done by recording the heart’s electrical activity using either noninvasive or minimally invasive methods, such as body surface or intracardiac measurements. Mathematical and numerical models of cardiac electrophysiology can provide insight into how electrical measurements can be used to detect ischemias. In the long run, the goal is to combine boundary measurements of potentials with a mathematical model of the heart’s electrical activity to identify the position, shape, and size of ischemias and/or infarctions. Ischemic regions can be modeled as electrical insulators using the monodomain model, which is a semilinear reaction-diffusion system that describes cardiac electrical activity comprehensively.

In this talk, I will focus on the case of an insulated heart without coupling to the torso presenting some results obtained recently in the case of the stationary and time-dependent monodomain model.

Anna Mazzucato (Penn State)

Title: Direct and Inverse problems in monitoring of faults

Abstract: I will discuss a model of elastic dislocations applicable to buried faults in the Earth’s crust in between seismic events. The forward problem  amounts to solving a non-standard transmission problem for a system of linear PDES in elastostatics, knowing the fault and how much the rock has slipped at the fault. The inverse problem consists in determining the geometry of the fault and the slip at the fault from surface measurements, which can be obtained from  GPS and satellite data. While the direct problem is well posed, the inverse problem is generally ill-posed unless assumptions are made on the fault.

I will present a uniqueness result for the inverse problem and an iterative reconstruction algorithm based on a distributed shape derivative, which measures the change in the rock displacement under infinitesimal movements of the fault and the slip. I will close with some simple numerical  tests from synthetic data.

This is joint work with Andrea Aspri (University of Milan), Elena Beretta (NYU-Abu Dhabi), and Maarten de Hoop (Rice University).

Fabio Cavalletti (U Milano)

Title: An Optimal Transport approach to Lorentzian Ricci bounds with applications

Abstract: 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 thought as "non-smooth Riemannian manifolds”.  Since the geometric framework of general relativity is the one of Lorentzian manifolds and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, it is natural to expect that optimal transport tools can be useful also in this setting. Additionally, while Einstein’s theory of gravity is formulated in a smooth setting, in many physical situations this smoothness must eventually breakdown suggesting to look for a non-smooth theory of Lorentzian manifolds. 

The aim of the talk is to introduce the topic and to report on recent progress: after recalling the general setting of Lorentzian pre-length spaces introduced by Kunzinger-Sämann, after Kronheimer-Penrose, we will discuss some basics of Lorentzian optimal transport theory and we define "timelike Ricci curvature and dimension bounds” for a possibly non-smooth Lorentzian space using entropic convexity along geodesics of probability measures.

Finally, we will discuss a new isoperimetric-type inequalities in Lorentzian signature. 

Based on joint works with A. Mondino.

Luca Lombardini (TU Wien)

Title: A fractional Cahn-Hilliard system

Abstract: In this talk, we introduce a fractional variant of the Cahn-Hilliard system. We first focus on the model case of the fractional Laplacian with homogeneous Dirichlet boundary conditions, and briefly show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as “minimizing-movements scheme”, which fits naturally with the gradient-flow structure of the equation.

The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, it can be applied to show existence of a weak solution also for a more general class of integro-differential operators, not necessarily linear or symmetric. These include, e.g., fractional versions of the p-Laplacian. Moreover, by adapting the argument to the case of regional fractional operators, we can prove the existence of solutions also in the interesting case of fractional Neumann boundary conditions. These aspects will be the object of the second part of the talk.

This is a joint work with E. Davoli and C. Gavioli.

Alessandro Audrito (PoliTO)

Title: Elliptic regularization of some parabolic semilinear free boundary problems

Abstract: In this talk I will present some recent results and some ongoing research about the elliptic regularization of a class of parabolic semilinear free boundary problems. All the results have been obtained in collaboration with Tomas Sanz-Perela (UB).

Louis Dupaigne (Université Lyon 1 Claude Bernard)

Title: Computing the best constant in Poincaré's inequality

Abstract: The optimal constant in Poincaré's inequality is often difficult to compute and only few examples are known. Gaussian space and the round sphere S^d are well-known examples. In fact, the former is a limiting case of the latter as d\to+\infty. I will present some new examples, which are conformal to a standard right cone embedded in Euclidean space, the base of which need not be circular and the volume of which is weighted. The interplay between the opening of the cone and the size of the weight at its vertex turns out to be central for the computation. Other intriguing phenomena appear when the weight is not homogeneous in every direction. 

Joint work with I Gentil, N Simonov and S Zugmeyer.

Alessio Martini (PoliTo)

Title: Sharp Lp estimates for the sub-Riemannian wave equation

Abstract: The Miyachi-Peral fixed-time Lp estimates with loss of derivatives are among the classical results for the Euclidean wave equation. It is well known that analogues of those estimates hold true for wave equations driven by more general elliptic operators than the standard Laplacian. Notably, a celebrated theorem by Seeger, Sogge and Stein on Fourier integral operators yields such estimates for the Laplace-Beltrami operator on any compact Riemannian manifold. In comparison, the wave equation driven by a (non-elliptic) sub-Laplacian on a sub-Riemannian manifold is much less understood, and sharp Lp estimates are known only in few particular cases. We report on recent progress based on joint work with Detlef Müller.

Massimiliano Morini (U Parma)

Title: A distributional approach to nonlocal curvature motions

Abstract: After reviewing the new distributional approach recently developed to provide a well-posed formulation of the crystalline mean curvature flow, we show how to extend it to some nonlocal motions. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the (n-1)-dimensional Minkowski pre-content.

Ilaria Fragalà (PoliMi)

Title: The geometric size of the fundamental gap

Abstract: The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in the N-dimensional Euclidean space. The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. In this talk I will discuss some very recent results which answer affirmatively such question and, going beyond, strengthen Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The results are contained in a paper in collaboration with Vincenzo Amato and Dorin Bucur.

Camillo De Lellis (IAS Princeton) [CANCELLED]

Title: Area-minimizing currents mod an integer

Abstract: Currents mod p are a suitable generalization of classical chains mod p, i.e. of finite combinations of smooth submanifolds with coefficients in the cyclic group $\mathbb Z_p$. By the pioneering work of Federer and Fleming it is possible to minimize the area in this context and, for instance, represent mod p homology classes with area minimizers. For $p>2$ typically (i.e. away from a small set of exceptional points) one would expect such minimizers to be a union of smooth minimal surfaces joining together (``in multiples of $p$'') at some common boundary. This is however surprisingly challenging to prove, especially for even $p$'s, and up until recently only known for $p=3$ and $4$ in codimension $1$. In this talk I will explain the outcome of a series of more recent works (some joint of the speaker with Hirsch, Marchese, Stuvard and Spolaor, some by Wickramasekera and Minter-Wickramasekera, and some joint of the speaker with Minter and Skorobogatova) which confirms this picture, with varying degrees of precision in a variety of situations.

Angela Alberico (IAC-CNR)

Title: Embedding theorems for fractional Orlicz-Sobolev spaces

Abstract: The optimal target space is exhibited for embeddings of fractional- order Orlicz-Sobolev spaces. Both the subcritical and the supercritical regimes are considered. In the former case, the smallest possible Orlicz target space is detected. In the latter, the relevant Orlicz-Sobolev spaces are shown to be embedded into the space of bounded continuous functions in Rn. Moreover, their optimal modulus of continuity is exhibited.

These results are the subject of a series of joint papers with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

Peter Sjögren (University of Gothenburg)

Title: Sharp estimates of heat kernels in the Jacobi and other settings 

Abstract: The main result presented is a sharp estimate for the Jacobi heat kernel, valid for all possible values of the parameters. Only a rough description of the rather technical proof will be given. We also indicate how this heat kernel relates to the heat kernel of a sphere or a compact rank-one symmetric space, or that of other settings.

This is joint work with A. Nowak and T. Z. Szarek. 

Maria Vallarino (PoliTo)

Title: Dyadic sets and dyadic Calderón–Zygmund Theory

Abstract: In this talk we will start recalling the definition and the main properties of the family of classical dyadic cubes in the Euclidean space ℝd, considering the dyadic maximal Hardy–Littlewood operator and the dyadic Calderón–Zygmund theory in this setting (see [4]). Thanks to a seminal paper by Christ [1], it is known that a family of dyadic sets can be constructed in doubling measure metric spaces, and a corresponding dyadic Calderón–Zygmund theory can be considered. 

Recently López-Sánchez, Martell and Parcet [5] developed a dyadic Calderón–Zygmund theory in more general measure metric spaces, possibly nondoubling. We shall first give the main ideas behind this theory, and then we will show how to apply it in the setting of nonhomogeneous trees with unbounded geometry, and deduce boundedness properties of Bergman projectors in this setting. This is based on a joint work with Conde Alonso, De Mari, Monti, Rizzo [2, 3]. 

References:

1. Christ, M., A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., Colloquium Mathematicum 60/61 (1990), 601–628, https://doi.org/10.4064/cm-60-61-2-601-628.

2. Conde Alonso J.M., De Mari, F., Monti, M., Rizzo E., Vallarino, M., Harmonic Bergman Projectors on Nonhomogeneous Trees, preprint.

3. De Mari, F., Monti, M., Vallarino, M., Harmonic Bergman Projectors on Homogeneous Trees, Potential Anal. (2023), https://doi.org/10.1007/s11118-023-10106-4.

4. Grafakos L., Classical Fourier Analysis, Springer (2016).

5. López-Sánchez, L.D., Martell, J.M., Parcet, J., Dyadic harmonic analysis beyond doubling measures, Adv. Math. 267 (2014), 44–93, https://doi.org/10.1016/j.aim.2014.08.001.

Bozhidar Velichkov (U Pisa)

Title: Regularity up to the boundary for optimal partition problems

Abstract: This talk is based on a recent joint work with Roberto Ognibene on the problem of optimal partition of a fixed domain (a box) with respect to the sum of the principal eigenvalues, where we prove regularity results for the free interface up to fixed boundary; in particular, we show that the subset of points of minimal frequency is regular and that the interior free interface is approaching the boundary orthogonally in a smooth way.

Serena Dipierro (UWAust)

Title: A strict maximum principle for nonlocal minimal surfaces

Abstract: Suppose that two nonlocal minimal surfaces are included one into the other

and touch at a point. Then, they must coincide. But this is perhaps less obvious than

what it seems at first glance.