I seminari "PDE a tutto SBAI" si svolgono nell'Aula 1B1 (RM002,  ex Palazzina B Via A.Scarpa 16) con cadenza bisettimanale,  il Giovedì alle ore 14,30. 

Organizzatori:

Massimo Grossi, Luisa Moschini, Francescantonio Oliva e Francesco Petitta  

Seminari 2023/2024

Prossimo Seminario 

23 Maggio 2024  -  Gabrielle Nornberg   (Universidad de Chile)

14.30

Fundamental solutions and Liouville results for nonlinear nonlocal operators in cones 

In this talk we discuss the existence of fundamental solutions for a class of nonlinear nonlocal uniformly elliptic operators defined in cones and its application to derive Liouville results for Lane-Emden nonlocal type equations.

Joint work with Disson dos Prazeres (Federal University of Sergipe, Brazil) and Alexander Quaas (Federico Santa María Technical University, Chile). 

Seminari Futuri 

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Seminari Passati

 09 Maggio 2024  -  Alberto Saldaña  (Universidad Nacional Autonoma de México)

14.30

Asymptotic analysis of Lane-Emden systems

Lawrence C. Evans once wrote: “One important principle of mathematics is that extreme cases reveal interesting structure.”  In this talk, we put this piece of mathematical wisdom to the test.  

First, we recall some known results on elliptic problems with power nonlinearity as the exponent tends to infinity.  Then we consider the case of the Lane-Emden system and study the profile of least-energy solutions as one of the exponents tends to infinity. In this case, the limit profile can be characterized as a least-energy solution of a p-biharmonic nonlinear equation which can be studied with tools from nonsmooth analysis.

These results were obtained in collaboration with Nicola Abatangelo (Università di Bologna) and Hugo Tavares (Universidade de Lisboa). 

 23 Aprile 2024  -  Elide Terraneo (Università degli Studi di Milano Statale)

14.30

Singular solutions to semilinear elliptic equations with exponential nonlinearities in 2-dimensions

By introducing a new classification of the growth rate of exponential functions, singular solutions for -\Delta u=f(u) in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity ``close" to f, which admits an explicit singular solution. Then, one obtains an approximate singular solution, and concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way, e.g., f(u) = u^re^{u^q}\ (q>1,r\in \mathbb{R}), 

f(u) =e^{u^{q}+u^r}\ ({q>1,\ q/2>r>0}) or f(u) = e^{e^u}. As a special case, our result contains a pioneering contribution by Ibrahim--Kikuchi--Nakanishi--Wei for u(e^{u^2}-1). 

Joint work with Yohei Fujishima (Shizuoka University, Japan),  Norisuke Ioku (Tohoku University, Japan) and Bernhard Ruf (Istituto Lombardo, Italy).

 21 Marzo 2024  -  Francesca Gladiali (Università degli Studi di Sassari)

14.30

On the critical points of solutions of Pdes in non-convex settings: the case of concentrating solutions

In questo seminario parlerò del numero dei punti critici delle soluzioni di Pdes ellittiche nonlineari in un dominio di R^2  non convesso contrattile oppure no.Proverò una stima sul numero dei punti critici per le soluzioni del problema di Gel’fand che si concentrano. Farò vedere che in alcuni casi la stima è ottimale. I risultati si basano su un lavoro in collaborazione con M. Grossi. 

 13 Marzo 2024  -  Alberto Enciso (ICMAT, Madrid)

14.00

The fearful symmetry of Neumann eigenfunctions 

We will discuss some recent results and open problems on the spectral geometry of planar domains that exhibit certain geometric properties. We will be mostly interested in how the “roundness” of the domain may have a qualitative impact (or not) on the structure of the eigenfunctions. Simple geometric ideas will be emphasized throughout. 

 7 Marzo 2024  -  Andrea Bisterzo (Sapienza, Università di Roma)

14.30

 Weak maximum principles for elliptic operators on unbounded Riemannian domains and an application to a symmetry problem

The necessity of a maximum principle arises naturally when one is interested in studying qualitative properties of solutions to partial differential equations. Generally, to ensure the validity of such principles, additional assumptions on the ambient space or on the differential operator need to be considered. The talk aims to address, using both of theseapproaches, the problem of proving (weak) maximum principles for second-order elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. At the end of the talk, we will see how to apply these maximum principles to recover a symmetry result for stable solutions to semilinear PDEs in isoparametric Riemannian domains. 

22 Febbraio 2024

14.30

 Antonio J. Martinez Aparicio (Universidad de Almeria)

Semilinear elliptic eigenvalue problems with infinitely many positive solutions

We study the behavior of the set of solutions of the semilinear elliptic problem -Δu = λf(u) in a bounded N-dimensional open set with Dirichlet boundary conditions. Here, f is a nonnegative continuous real function with multiple zeros. First, we analyze the set of the solutions whose maximum is between two consecutive positive zeros of f, arriving to the existence of an unbounded continuum of solutions with C-shape. Then, we study the asymptotic behavior of the countable many unbounded continua in the case in which f has a sequence of positive zeros, proving for some model cases that every λ>0 is a bifurcation point (either from infinity or from zero) that is not a branching point.

08 Febbraio 2024

14.30

 Delia Schiera  (Universidade de Lisboa)

Maximum principles and related problems for a class of nonlocal extremal operators 

I will consider a class of degenerate nonlinear operators that are extremal among operators with one dimensional fractional diffusion and that approximate the so-called truncated Laplacians. I will show some properties of these operators, emphasizing the differences both with the local equivalent operators and with more standard nonlocal operators such as the fractional Laplacian. In particular, continuity properties, validity of comparison and maximum principles, and their relation with principal eigenvalues, will be presented. Joint work with Isabeau Birindelli and Giulio Galise.

25 Gennaio 2024

14.30

Davide Buoso (Università degli Studi del Piemonte Orientale “A. Avogadro”)

Bulk-boundary eigenvalue problems: a review

Diffusion equations in their linearized versions naturally lead to the study of elliptic eigenvalue problems where the eigenvalue appears both in the interior and on the boundary. This kind of spectral problems have surged in the literature only recently and give rise to many appealing questions. In this talk we will review two such problems that showed to be particularly interesting: the first one comes from the Allen-Cahn equation and is associated with the Laplacian, and has already been used to prove interesting results in spectral geometry. The second one comes from the Cahn-Hillard equation and is associated with the bilaplacian. After introducing these two problems, we will recall some basic facts from Spectral Theory and Spectral Geometry to provide context, and then discuss major properties and open problems concerning their eigenvalues. 

11 Gennaio 2024

14.30

Matheus Frederico Stapenhorst (Universidade Estadual Paulista, Sao Paulo, Brazil)

A class of singular elliptic problems in the plane 

 In this seminar we consider a class of singular elliptic problems defined in a bounded region of the plane with zero Dirichlet boundary condition. More precisely, we discuss the solvability of problems of the form -Laplacian of u + positive singular term = f(u), where f is considered to be of subcritical or critical growth with respect to the Trudinger-Moser inequality. Our approach is as follows: we first use variational methods to obtain solutions of suitable approximating problems. Next, we study the regularity of these solutions and conclude that the limit is a solution of the original problem in a weak sense. This is a part of the author's PhD Thesis, advised by Prof. Marcelo Montenegro at Universidade Estadual de Campinas, Brazil.

14 Dicembre 2023

14.30

Daniele Castorina (Università degli studi di Napoli Federico II)

Mean field sparse optimal control of systems with additive white noise

We analyze the problem of controlling a multi-agent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders). For such a controlled system with a SDE constraint, we introduce a rigorous limit process towards an infinite dimensional optimal control problem constrained by the coupling of a system of ODE for the leaders with a McKean-Vlasov-type SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the Γ-limit of the cost functionals for the finite dimensional problems. Joint work with Giacomo Ascione (SSM Napoli) and Francesco Solombrino (Napoli Federico II). 

30 Novembre 2023

14.30

Sergio Cruz Blázquez (Universidad de Granada)

Infinite Morse Index Solutions to the Fractional Yamabe Problem

In this talk we are interested in the qualitative properties of the solutions to the fractional Yamabe problem in R^n which present an isolated singularity. In particular, we prove that the Morse index of any such solution is infinity. The proof uses an Emden-Fowler type transformation, so that we can pass to a nonlocal 1D problem posed in R. The main reference is

[1] S. C., A. De la Torre and D. Ruiz, Qualitative Properties for Solutions of the Fractional Yamabe Problem, to appear in Discrete and Continuous Dynamical Systems (2023). Arxiv version: https://arxiv.org/abs/2207.09886

16 Novembre 2023

14.30

Antonio J. Fernandez (Universidad Autónoma de Madrid)

A Schiffer-type problem with applications to stationary Euler flows 

If on a smooth bounded domain $\Omega\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $\Omega$ be a disk or an annulus? This question can be understood as a weaker analog of the celebrated Schiffer conjecture, in that the function $u$ is here allowed to take a different constant value on each connected component of $\partial \Omega$ yet many of the known rigidity properties of the original problem are essentially preserved. In this talk we provide a negative answer by constructing a family of nontrivial doubly connected domains $\Omega$ with the above property. Then, we will show how our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. The talk is based on a joint work with Alberto Enciso, David Ruiz and Pieralberto Sicbaldi. 

12 Ottobre 2023

14.30

Benedetta Pellacci (Università della Campania ``Luigi Vanvitelli'')

Spectral Optimization problems arising in logistic models.

We will discuss some recent results concerning weighted eigenvalue problems in bounded Lipschitz domains, under Robin boundary conditions. This kind of optmization problems naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in an heterogeneous habitat. Looking for this optimal distribution amounts to minimize a principal eigenvalue with respect to the sign-changing weight. The optimization problem is completely solved in one dimension, and we will discuss some recent result in the presence of an anisotropic diffusion. The analogous study  is open in its generality in higher dimension and we will present some asymptotical results in the case of Neumann boundary conditions.

Joint works with Dario Mazzoleni (Università di Pavia), Giovanni Pisante (Università della Campania ``Luigi Vanvitelli''), Delia Schiera  (Universidade de Lisboa) and Gianmaria Verzini (Politecnico di Milano).

5 Ottobre 2023

14.30

Yuxia Guo (Tsinghua University, Beijing)

An elliptic problem with periodic boundary conditions involving critical growth.

We consider an elliptic problem involving critical growth in a strip, satisfying the periodic boundary condition. We first prove the existence of a single bubble solution. As a consequence, we obtain that the prescribed scalar curvature problem in R^N has solutions which are periodic in its first k variable (k<\frac{N-1}{2}) if the scalar curvature k(y) is periodic.

15.30

Shusen Yan (Central China Normal University, Wuhan)

Periodic solution for  Hamiltonian type systems with critical growth

We consider an elliptic system of Hamiltonian type in a strip in R^N, satisfying the periodic boundary condition

for the first k variables. In the superlinear case with critical growth, we prove the existence of a single bubbling solution for the system under an optimal condition on k.  The novelty of the paper is that all the estimates needed in the proof of the existence result can be obtained once the Green's function of the Laplacian operator in a strip with periodic boundary conditions is found.

28 Settembre 2023

Manuel del Pino (University of Bath) 

Gluing methods in the Water Wave Problem 

In the classical Water Wave Problem, we construct new overhanging solitary waves by a procedure resembling desingularization of the gluing of constant mean curvature surfaces by tiny catenoidal necks. The solutions here predicted have long been numerically detected. This is joint work with Juan Davila, Monica Musso, and Miles Wheeler.