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 2024/2025
Prossimo Seminario
11 Giugno 2025 - Pier Domenico Lamberti (Università di Padova)
14.30
Spectral stability of Maxwell’s equations in varying cavities
We consider the spectral problem for the stationary Maxwell's equations under homogeneous boundary conditions in a cavity, and we investigate how the eigenvalues depend on perturbations of the cavity's shape. The problem reduces to the analysis of the curl-curl operator. We study families of diffeomorphic domains and discuss the analytic dependence of the eigenvalues on the domain shape. In particular, we prove Hirakawa’s formula for the shape derivatives and derive a Rellich–Pohozaev-type identity.
Furthermore, we address a related shape optimization problem and show that the corresponding Faber–Krahn inequality does not hold. Time permitting, we also consider families of domains undergoing oscillatory boundary perturbations and establish spectral stability results under minimal assumptions on the oscillation strength. In particular, we highlight the role of suitable uniform Gaffney inequalities, which we prove for our purposes.
Seminari Futuri
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Seminari Passati
27 Marzo 2025 - Hans Knüpfer (U. Heidelberg)
14.30
2D Stokes Problem With a Moving Contact Line
We consider the evolution of a two–dimensional liquid droplet on a solid substrate in the presence of a contact point. At the liquid-solid interface we assume Navier slip. At the triple point we assume a constant contact angle, given by Young’s law. The main result is well–posedness and regularity of the solution of the corresponding linearized problem in a class of weighted Sobolev spaces.
This is a joint work with M. Bravin (U. Santander), M. Gnann (TU Delft), N. Masmoudi (NYU AD), F. Rodenburg (TU Delft), J. Sauer (U. Jena).
13 Marzo 2025 - Francesca Gladiali (Università di Sassari)
14.30
Solutions to general elliptic equations with many critical points on nearly geodesically convex domains
Given a complete d-dimensional Riemannian manifold (M, g) I will prove that, for any p ∈ M, any nonlinearity f(q, u) with f(p, 0) > 0 and for any integer n ≥ 2, there exists a sequence of smooth bounded domains Ω_k ⊂ M containing p and corresponding positive solutions u_k : Ω_k → R^+ to the Dirichlet boundary problem
−∆_g u_k = f(·, u_k) in Ω_k,
u_k = 0 on ∂Ω_k .
such that the solution u_k have exactly 2n − 1 nondegenerate critical points in Ω_k (specifically, n nondegenerate maxima and n − 1 nondegenerate saddles). Moreover the domains Ω_k are star-shaped with respect to p and become “nearly geodesically convex”, in a precise sense, as k → +∞. The proof relies on similar results in R^d, d ≥ 3, for the torsion problem.
The talk is based on past and ongoing results involving M. Grossi and A. Enciso.
24 Ottobre 2024 - Qing Guo (Minzu University of China)
14.30
Segregated solutions for nonlinear Schrödinger systems with sublinearly coupled nonlinearities
In this talk, we use an enhanced Lyapunov-Schmidt reduction method to study a specific class of nonlinear Schrödinger systems with sublinear coupling terms. We establish the existence of infinitely many nonnegative segregated solutions of the following system:
-∆u + K_1(x)u = μ|u|^(p-2)u + β|v|^(p/2)|u|^(p/2-2)u, x ∈ R^N
-∆v + K_2(x)v = ν|v|^(p-2)v + β|u|^(p/2)|v|^(p/2-2)v, x ∈ R^N
where p ∈ (2, 4), K_j(x), j = 1, 2, represent radially symmetric potential functions, μ > 0, ν > 0, and β<0 is the coupling coefficient.
Due to the sublinearity and nonsmooth nature of the coupling terms, traditional reduction methods face challenges. To address these, we propose a novel framework combining variational and perturbation techniques, using a refined inner-outer decomposition approach. Precisely, we construct solutions that concentrate at distinct points near infinity. These solutions also show a 'dead core' phenomenon, unique to this regime, where non-strict positivity is observed.
19 Settembre 2024 - Mirco Piccinini (Università di Parma)
14.30
Asymptotic approach to singular solutions for the CR Yamabe equation
We will investigate the effects of the lack of compactness in the critical Folland-Stein(-Sobolev) embedding in the Heisenberg group. In particular, by means of Γ-convergence techniques, we will show that optimal functions of a subcritical energy approximation of the Sobolev quotient in bounded (not necessarily regular) domains do concentrate energy at a single point. Moreover, in clear accordance with the underlying geometry, we show that the concentration of the energy happens at a critical point for the Robin function (i. e., the diagonal of the regular part of the Green function associated to the involved domain), thus proving that an analogous of a conjecture of Brezis and Peletier does still hold in the Heisenberg group.
Seminari 2023/2024
Seminari Passati
20 Giugno 2024 - Abdelrazek Dieb (Ibn Khaldoun University of Tiaret (Algeria))
14.30
Existence and non-existence results for non-linear elliptic systems involving Hardy potential
We discuss existence and non-existence results for non-negative super-solution to a class of gradients-potential systems with Hardy terms. In particular the existence of optimal critical curves that separates the existence and the non-existence regions is proved.
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).
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.