50 years of Mountain Pass Theorem

a conference to celebrate the theorem and its authors

November 23 – November 25, 2023, SISSA, Trieste

Program

Venue: Big meeting room, 7 floor, SISSA, via Bonomea, 265 - 34136 Trieste

Thursday, November 23

• 15:00: Opening addresses
• 15:30–16:30: David Arcoya, (Universidad de Granada): Passing through mountains
  It is summarized the results of joint papers with Antonio Ambrosetti (Differential and Integral equations, 33 (2020), 92-112) and with Caterina Sportelli (Z. Angew. Math. Phys. (2023) 74:91) where we prove a version of the Mountain Pass Theorem for relativistic problems with singular terms. This version is applied to study either the existence of periodic solutions for the relativistic Lorentz force equation \begin{equation*} \left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'=\vec{E}(t,q) + q'\times \vec{B}(t,q), \end{equation*} with singular electric field $\vec{E}(t,q)$ and smooth magnetic field $\vec{B}(t,q)$, or the relativistic pendulum \begin{equation*}\left( \frac{u'}{\sqrt{1-u'^2}} \right)'+F'(u)=0 \end{equation*} with singular nonlinearity $F'(u)$.
• 16:30–17:00: coffee break
• 17:00: Paul H. Rabinowitz, (University of Wisconsin, Madison). Some reminiscences of Antonio and the origins of the Mountain Pass Theorem.
• 17:30–18:30: Michael Struwe, (ETH-Zürich): Plateau flow, or the heat flow for half-harmonic maps
  Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N$ in $\mathbb{R}^n$, recently studied by Wettstein in the fractional setting, and for arbitrary finite-energy data we obtain a result fully analogous to my result for the harmonic map heat flow of surfaces from 1985, and in similar generality. When $N$ is a smoothly embedded, oriented closed curve, the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces. In particular, the results invite to study the parametric Plateau problem without a monotonicity requirement.

Friday, November 24

• 9:00–10:00: Roberta Musina, (Università di Udine): Existence of planar H-loops via Hardy's inequality and the Mountain Pass Theorem
  Given a continuous function on $\mathbb{R}^2$, we study the existence of non-constant, 2π-periodic solutions to the 2-dimensional Hamiltonian system $$u'' = |u'| \text{H}(u)iu'. \tag{P}$$ Any non-constant solution to (P) parametrizes a closed planar curve having prescribed curvature H at each point. Our interest in (P) is motivated also by its relations with Arnold’s problem on H-magnetic geodesics. Problem (P) admits a variational formulation; under reasonable assumptions on the prescribed (non-constant) curvature H, the associated energy functional has a nice Mountain-Pass geometry. However, due to the groups of dilations and translations in $\mathbb{R}^2$, the Palais-Smale condition fails to hold, and in fact there could exists unbounded Palais-Smale sequences. We will present an existence result which is strongly based on Hardy’s inequality for functions of two variables.
  This is joint work with Gabriele Cora (Università di Torino).
• 10:00–11:00: Ivar Ekeland, (Université Paris-Dauphine): The mountain pass theorem and periodic solutions of Hamiltonian systems
  I will give Ghoussoub's version of the mountain pass theorem, and I will apply it to finding periodic solutions of convex Hamiltonian systems
• 11:00–11:30: coffee break
• 11:30–12:30: David Ruiz, (Universidad de Granada): Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime
  The Gross-Pitaevskii equation is a Schrödinger equation under the effect of a Ginzburg-Landau potential, and has been proposed to study different phenomena like the Bose-Einstein condensation. In this talk we will focus on the existence of finite energy traveling waves for this equation. Some previous existence results have been obtained via a constrained minimization procedure. This has the advantage of providing orbital stability of the solutions but, as a drawback, the speed of propagation appears as a Lagrange multiplier and is not controlled. In this talk we present a general existence result for almost all subsonic speeds.
  This is joint work with Jacopo Bellazzini (U. Pisa).

14:15–17:00 Ambrosetti Medal Award Ceremony

• 15:00–16:00: Javier Gomez Serrano, (Brown University): Unveiling Singularities in Nonlinear PDE through Machine Learning
  The quest to understand and predict singularities in nonlinear partial differential equations has been a long-standing challenge in mathematical and computational sciences. In recent years with the development of new hardware and software, a new paradigm in physics and engineering has emerged via machine learning. In this talk I will present a possible way of interaction between the two worlds of mathematics and artificial intelligence, and how computer-assisted proofs may play a prominent role in the next few years in the context of mathematically proving rigorously the existence of finite time singularities. Specifically I will focus on different equations in fluid dynamics, but our methods are shown to be robust, generic and adaptable to many situations.
• 16:00–17:00: Alessandro Carlotto, (Università di Trento): Minimal surfaces: what future after the min-max revolution?
  The striking advances in the min-max theory for the area functional have led, over the past decade, to the solution of a variety of outstanding problems in geometry, including Yau’s conjecture ensuring the existence of infinitely many minimal hypersurfaces in any assigned compact ambient Riemannian manifold. In the case of ambient manifolds with boundary, one obtains relative cycles, namely free boundary minimal hypersurfaces. However, such methods concern very weak notions of ``surfaces'' and, correspondingly, build upon weak notions of convergence (namely: those peculiar of geometric measure theory): as a result, this sort of approach is not topologically effective and it is in general unreasonable to expect to have any control on the type of submanifolds it provides.
  Springing from such issues, I will try to describe what I envision for the post-revolutionary era by focusing on the simplest and visually most appealing case of free boundary minimal surfaces in the Euclidean ball. I will discuss how to construct such surfaces in a controlled fashion and how to distinguish them by virtue of the fine analysis of their (equivariant or absolute) Morse index. As a surprising byproduct, we will start to sketch a comparative picture of variational vs. perturbative methods, and indicate what is still to be understood in that direction.
• 17:00–17:30: coffee break
• 17:30–18:30: Silvia Cingolani, (Università degli Studi di Bari Aldo Moro): Nonlocal NLS equations: concentration around a saddle point in a degenerate setting
  In my talk I will present a new variational approach, for detecting solutions to local or nonlocal NLS equations, concentrating around a critical point, for instance a saddle point, in a degenerate setting, where finite dimensional reduction arguments fail to hold. In particular a shift deformation flow is generated in an augmented space to regain compactness properties.
  The seminar is based on joint papers with Kazunaga Tanaka (Waseda University, Tokyo).

Saturday, November 25

• 9:15–10:15: Beatrice Langella, (SISSA): Time periodic solutions of resonant Klein-Gordon equations: a variational approach
  In this talk I will focus on a class of completely resonant Klein-Gordon equations on the 3 dimensional sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, which arise as toy models in General Relativity. I will show that these equations admit small amplitude, time periodic solutions. Their existence is obtained by a variational Lyapunov-Schmidt decomposition, which reduces the problem to the search of Mountain Pass critical points of a restricted Euler-Lagrange action functional. In order to gain compactness of the gradient of such a functional and smoothness of the critical points, a key point is to implement Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on $\mathbb{S}^3$.
  Based on a joint work with Massimiliano Berti and Diego Silimbani.
• 10:15–10:30: coffee break
• 10:30–11:30: Gianmaria Verzini (Politecnico di Milano): Regularized variational principles for the perturbed Kepler problem
  We develop a method that combines the use of variational techniques with regularization methods to study the periodic problem associated to the perturbed Kepler system \[ \ddot x = -\frac{x}{|x|^3} + p(t), \quad x=x(t) \in \mathbb{R}^d, \] where $p:\mathbb{R}\to\mathbb{R}^d$ is smooth and $T$-periodic, $T>0$. The existence of critical points for the action functional associated to the problem is proved via a non-local change of variables inspired by Levi-Civita and Kustaanheimo-Stiefel techniques. As an application, we prove that the perturbed Kepler problem has infinitely many generalized $T$-periodic solutions for $d=2$ and $d=3$, without any symmetry assumptions on $p$.
  This is a joint work with Vivina Barutello (Torino) and Rafael Ortega (Granada).

Partecipation is free, but you must register by sending an email to vittorio.cotizelati at unina.it

Scientific Committee: Massimiliano Berti, Vittorio Coti Zelati, Andrea Malchiodi, Filomena Pacella, Paul H. Rabinowitz