Abstracts Fall Semester 2024

Title: Sard properties for polynomial maps in infinite dimension with applications to the Sard conjecture in Carnot groups

Abstract: Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional the property may fail. I will present a recent work with A. Lerario and D. Tiberio (SISSA), establishing sharp quantitative criteria for the validity of Sard’s theorem in this setting. Our motivation comes from sub-Riemannian geometry and we provide in particular applications of our results to the Sard conjecture on Carnot groups.

Title: Inradius Vs. Poincaré

Abstract: We discuss some estimates on the sharp $L^p$ Poincar\'e constant of an open set, in terms of various notions of inradius. We start by reviewing some classical two-dimensional results by J. Hersch, E. Makai and W. K. Hayman, together with their extensions. We then present a more general definition of inradius and show that this can be used to completely characterize the validity of the $L^p$ Poincar\'e inequality. Such a characterization comes with a two-sided estimate of the sharp constant, which extends and completes a theorem by V. Maz'ya.

Some of the results presented have been obtained in collaboration with the Ph.D. student Francesco Bozzola.

Title: Concentration and oscillation analysis of semilinear elliptic equations with exponential growth in a disc

Abstract: We investigate blow-up positive solutions of semilinear elliptic equations with exponential growth. Especially, we are interested in concentration and oscillation phenomena on supercritical problems. We first detect infinite sequences of bubbles by a scaling technique. The precise description of each bubble is completed via a limit equation with mass recurrence formulas. This enables us to proceed to our second main discussion on oscillation phenomena. A key observation is that the infinite sequences of bubbles cause infinite oscillations, around singular solutions, of blow-up solutions. Thanks to this, we finally arrive at a proof of infinite oscillations of bifurcation diagrams which yield the existence of infinitely many solutions.

Title: Weighted Hardy-Sobolev Type Inequalities with Remainder 

Abstract: This talk focuses on indefinite quasilinear elliptic problems involving weighted terms on unbounded domains, potentially with unbounded boundaries. We will explore existence results using variational methods applied to weighted function spaces and present several Liouville-type results for this class of problems. 

Title: Coupled Elliptic systems with sublinear growth 

Abstract: Consider the coupled elliptic system

−∆u+ u= ρ_1(x)u^{p_1} + λv in R^N

−∆v+ v= ρ_2(x)v^{p_2} + λu in R^N,

u(x),v(x) →0 as |x|→∞.

We observe that in 2008, A. Ambrosetti, G. Cerami and D. Ruiz proved the existence of positive bound and ground states in the case λ∈(0,1), p_1 = p= p_2, 1 <p<2*−1, ρ_1(x) and ρ_2(x) tend to one at infinity. In this work we complement their result, because

we show that the previous system has no solutions when 0 <p_1,p_2 <1, as well as we establish sharp hypotheses on the powers 0 <p_1,p_2 , the parameter λ and the weights ρ_1(x), ρ_2(x) that will allow us to obtain the existence and uniqueness of a positive bounded solution.

Title: New Spectral Generalizations of the Bishop–Gromov Volume Inequality and Bonnet–Myers Diameter Estimate

Abstract: It is classical that if an n-dimensional smooth complete manifold M without boundary satisfies Ric≥n−1, then it is compact, its volume is bounded above by that of the round unit sphere, and the fundamental group of M is finite.

In this talk, I will discuss a new sharp and rigid generalization of this result. Let Δ denote the Laplace operator on functions. I will prove that if, on a compact n-manifold, the first eigenvalue of the Schrödinger operator −(n−2)/(n−1)Δ+Ric is greater than or equal to n−1, then the same conclusions as above hold.

I will address the sharpness of this result, rigidity, and a spectral version of the Bonnet–Myers diameter estimate too. Time permitting, I will also discuss applications of this result to geometric problems, such as the stable Bernstein problem.

Title: Regularity in diffusion models with gradient activation

Abstract:  In this talk, we discuss sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient surpasses a given threshold. This is joint work with Aelson Sobral, King Abdullah University of Science and Technology, KAUST - Saudi Arabia, and Eduardo Teixeira, University of Central Florida - EUA.

Title: Pointwise Convergence to Initial Data of Some Evolution Equations

Abstract: Consider the Cauchy problem for the fractional heat equation or the Caffarelli–Silvestre extension problem on a manifold. Under what conditions do solutions converge pointwise a.e. to the initial data as t → 0? We will discuss recent results concerning measures ν for which this occurs whenever the initial data are in Lp(ν), 1 ≤ p ≤ ∞. 

The talk will be based on joint work with E. Papageorgiou (Paderborn) 

Title: Optimal control and early states reconstruction in phase-field models for tumor growth

Abstract: In this talk we present recent results on well-posedness, optimal control and inverse identification of the initial condition for two tumor growth models related to prostate and brain tumors. 

The first model is described by a PDE system ruling the evolution of the tumor phase parameter, the nutrient and the prostate index and it consists of a coupling between an Allen-Cahn equation and two reaction diffusion equations. The system is subjected to combined cytotoxic and antiangiogenic therapies, and we propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics.  We also illustrate the inverse identification of the initial data, starting from a measurement at the final time. This can be useful in medical applications, after some diagnostic images are obtained, to locate with more precision the areas where the tumour started growing, as well as some information on the initial distribution of nutrient and PSA, which could still give more insight to the practitioners. These pieces of information can then be used to better calibrate therapies on the patients. The last problem is finally investigated for a different PDE system coupling a Cahn-Hilliard type equation for the tumor phase and a reaction-diffusion equation for the nutrient proportion describing a brain tumor type model. 

These are joint projects with A. Agosti, E. Beretta, C. Cavaterra. P. Colli, M. Fornoni, G. Lorenzo, A. Reali. 

Title: Higher dimensional worm domains

Abstract: In 1977, Diederich and Fornaess constructed a family of domains in two-dimensional complex space, called "worm domains", that turned out to be counterexamples to a number of important phenomena in several complex variables. Since the 90s, it has been clear that worm domains play a key role in our (still incomplete) understanding of the global regularity theory of the d-bar Neumann problem. I will report on joint work (arXiv 2410.08736) with S. Calamai (Università degli Studi di Firenze), where we show how to construct a class of higher dimensional "worm domains". I will also raise a number of open questions and speculate about possible further progress in the global regularity theory on worm domains. 

Title: Partial regularity in nonlocal problems

Abstract: The theory of partial regular regularity for elliptic systems replaces the classical De Giorgi-Nash-Moser one for scalar equations asserting that solutions are regular outside a negligible closed subset called the singular set. Eventually, Hausdorff dimension estimates on such a set can be given. The singular set is in general non-empty. The theory is classical, started by Giusti & Miranda and Morrey, in turn relying on De Giorgi's seminal ideas for minimal surfaces. I shall present a few results aimed at extending the classical, local partial regularity theory to nonlinear integrodifferential systems and to provide a few basic, general tools in order to prove so called epsilon-regularity theorems in general non-local settings. From recent, joint work with Cristiana De Filippis (Parma) and Simon Nowak (Bielefeld). 

Title: The Gurtin–Pipkin heat equation: Old and new results

Abstract: We consider an abstract version of the integrodifferential equation modeling hereditary heat conduction of Gurtin–Pipkin type. Under suitable albeit quite general assumptions on the convolution kernel, the equation generates a contraction semigroup acting on a certain Hilbert space. Although the decay properties of the semigroup are nowadays well understood, several important issues related to the structure of the spectrum of its infinitesimal generator have not yet been investigated. In this talk, we provide some answers in that direction, demonstrating in particular the impossibility to have arbitrarily fast decays. For the most relevant physical case of the exponential kernel, we will also prove that the semigroup fulfills the so-called spectrum determined growth condition, telling that the decay type is fully dictated by the spectrum of the generator. In some cases, the optimal decay rate turns out to be actually attained.

Title: Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Abstract: Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.

Based on joint work arXiv:2303.00398 (J École Poly 11:954-1010, 2024) joint work with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham).

Title: On a heat flow for half harmonic maps

Abstract: In this seminar I will discuss some existence, via a Ginzburg-Landau approximation

scheme, and (partial)-regularity results for the solutions of an (unusual) heat flow

for half harmonic maps. This flow satisfies a monotonicity formula and is intimately

related with the so-called heat flow of harmonic maps with free boundary. 

This is a joint work with A. Hyder, Y. Sire and C. Wang.

Title: A numerical method for the solution of boundary value problems on Lipschitz planar domains

Abstract: The Unified Transform Method (UTM) was pioneered in the early ’90s by A. S. Fokas and I. M. Gel’fand in their study of the numerical solution of boundary value problems for elliptic PDEs and for a large class of nonlinear partial differential equations (PDEs). The UTM provides a connection between the Fourier Transform method (FT) for linear PDEs and its nonlinear counterpart, namely the Inverse Spectral method – also known as Non Linear Fourier Transform method (NLFT). At the heart of the matter is a new derivation of the FT method for linear equations in one and two (space) variables that follows the same conceptual steps needed to implement the NLFT method for a class of nonlinear evolution equations, thus pointing to a unified approach to the numerical solution of linear and nonlinear PDEs. 

From the very beginning, the UTM has attracted a great deal of interest in the applied mathematics community. A multitude of versions of the original method have since been developed, each dealing with a specific family of equations. Here we focus on a 2003 result of A.S. Fokas and A.A. Kapaev pertaining to the study of boundary value problems for the Laplacian on convex polygons: their original approach relied on a variety of tools (spectral analysis of a parameter-dependent ODE; Riemann-Hilbert techniques, etc.) but it was later observed by D. Crowdy that the method can be recast within a complex functiontheoretic framework (the Cauchy integral) which, in turn, extends the applicability to so-called circular domains, namely domains bounded by arcs of circles (with line segments being a special case). 

We extend the original approach of Fokas and Kapaev for polygons, to arbitrary convex domains. It turns out that ellipses (which are not circular in the sense of Crowdy) are of particular relevance in applications to engineering because the most popular heat exchangers (the shell-and-tube exchangers) have elliptical cross section. In this talk I will describe a complex function-theory based new algorithm for convex domains, and will highlight the numerical challenges that arise when implementing it. 

Time permitting, I will describe some very recent new results where further tools from complex analysis (the Szegő projection and conformal mapping) allow the application of the above algorithm also to non-convex domain shapes. 

This is joint work with J. Hulse (Syracuse University), S. Llewellyn Smith (UCSD & Scripps Institute of Oceanography) and Elena Luca (The Cyprus Institute).