Applied Analysis and PDEs Seminar

June 16, 2023
Georgios Domazakis (University of Sussex)
Rigidity of extremals for perimeter inequality under Schwarz symmetrisation
Abstract. This talk will discuss the rigidity of extremals for perimeter inequality under Schwarz symmetrisation. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the Schwarz symmetric set. We will present sufficient and necessary conditions for rigidity under this framework. Our analysis will based on the properties of the corresponding distribution function, and on a careful study of the transformations that can be applied on the symmetric set, without creating any perimeter contribution.
Webex link (3:00 PM - 4:00 PM, Friday, June 2, 2023)

June 2, 2023
Michael Sigal (University of Toronto)
Ginzburg-Landau equation on non-compact Riemann surfaces
Abstract. In this talk I will consider Ginzburg-Landau equations (GLE) on non-compact surfaces, more precisely on line bundles over such surfaces. GLE is a system of two equations for section and connection on these bundles yielding a crude model for a quantum engine where electrons are coming from infinity through one of the cusps and then departing through another. In this talk I will report on recent results with Nick Ercolani and Jingxuan Zhang on existence of energy minimizing solutions for such equations. In the talk I will introduce all necessary definitions beyond differential forms and exterior derivatives. In particular, no preliminary knowledge of Riemann surfaces and line bundles will be requiref.
Webex link (3:00 PM - 4:00 PM, Friday, June 2, 2023)

May 19, 2023
Filippo Santambroggio (Institut Camille Jordan, Université Claude Bernard - Lyon 1)
Optimal trajectories in L1 and under L1 penalizations
Abstract. I will present a recent work (just posted online ) in collaboration with my student Annette Dumas. Motivated by a MFG model where the trajectories of the agents are piecewise constants and agents pay for the number of jumps, we study a variational problem for curves of measures where the cost includes the length of the curve measures with the L1 distance, as well as other, non-autonomous, cost terms arising from congestion effects. We prove several regularity results (first in time, then in space) on the solution, based on suitable approximation and maximum principle techniques. Modern algorithms in non-smooth convex optimization allow to numerically simulate such solutions.
Webex link (3:00 PM - 4:00 PM, Friday, May 19, 2023)

May 5, 2023
Dimitrios Gazoulis (University of Crete)
P-functions and their Applications to Nonlinear Equations
Abstract. We introduce the notion of P-function for fully-nonlinear equations and study some applications. P-functions can be thought as quantities of a function u and its higher order derivatives that are related to a differential equation and have the property that satisfy the maximum principle. We study P-functions for a class of quasilinear elliptic equations and provide some general criterion for obtaining such quantities. In addition, we prove some gradient bounds and De Giorgi-type properties of entire solutions. As a special case, we obtain a gradient bound for the Allen-Cahn equations that differs from the Modica inequality.
Webex link (3:00 PM - 4:00 PM, Friday, May 5, 2023)

March 31, 2023
Ioannis Anapolitanos (Karlsruher Institut für Technologie)
How molecules get in shape
Abstract. Chemical properties of a molecule are determined by its shape. For example, the energy needed to initiate chemical reactions depends highly on the shape and structure of the involved molecules. It has also been known for a long time that the biological properties of protein molecules are determined by their structure. This naturally leads to several deep questions: How to determine the structure of molecules, is there a way to optimize the energy needed for chemical reactions, and so on. I will explain how to mathematically formulate this type of question for a specific group of chemical reactions, which are called isomerizations. This involves a single molecule, which changes its shape. Often, the initial and final shape of a molecule are given and one is seeking a path in the configuration space of the molecule, which describes how it changes its shape. The calculation of such reaction paths is very standard in quantum chemistry. While in quantum chemistry it is tentatively assumed that an optimal reaction path exists, this is far from obvious from a mathematical point of view. In fact, it is still an open problem! I will discuss recent results which provide a first step in this direction. In particular, I will describe some fundamental properties of molecules, concerning their structure and the forces between them. While our personal experience of the world involves mostly classical terms, the properties of molecules and matter at a microscopic scale necessarily involve quantum mechanics. However, for the purpose of this talk, I will not assume any previous knowledge of quantum mechanics. The talk is based on two recent papers, one with Mathieu Lewin, which is published, the other one being work in arxiv with Marco Olivieri and Sylvain Zalc.
Webex link (3:00 PM - 4:00 PM, Friday, March 31, 2023)

December 15, 2022
Konstantinos Gkikas (University of Athens)
Poisson problems involving fractional Hardy operators and measures
Abstract. We study the Poisson problem involving a fractional Hardy operator and a measure source. The complex interplay between the nonlocal nature of the operator, the peculiar effect of the singular potential and the measure source induces several new fundamental difficulties in comparison with the local case. To overcome these difficulties, we perform a careful analysis of the dual operator in the weighted distributional sense and establish fine properties of the associated function spaces, which in turn allow us to formulate the Poisson problem in an appropriate framework. In light of the close connection between the Poisson problem and its dual problem, we are able to establish various aspects of the theory for the Poisson problem including the solvability, a priori estimates, variants of Kato's inequality and regularity results (Joint work with H. Chen and P.T. Nguyen). K. G. acknowledges the support by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project Number: 59)

December 8, 2022
Zhiyuan Geng (Basque Center for Applied Mathematics, Spain)
On the triple junction problem without symmetry hypotheses
Abstract. In this talk, I will present some joint work with Nicholas Alikakos on the vector Allen-Cahn equation with a triple-well potential. Without assuming any symmetry hypothesis on the solution, we construct an entire minimizing solution that possesses a weak triple junction structure. A tight energy lower bound is established, which is crucial for estimates of the location and size of the diffuse interface. Other key tools used in our analysis include the 1D heteroclinic connection, Caffarelli-Córdoba density estimates and a blow-up analysis near the diffuse interface.

December 1, 2022
Giorgio Fusco (University of L'Aquila, Italy)
On the existence of triple junctions in bounded domains
Abstract. We consider a smooth nonnegative potential with exactly three nondegenerate zeros. A model of the bulk energy of a material which can exist in three different phases. By a variational approach we show that the connectivity of the phases is a sufficient condition for the existence of a triple junction solution of the vector Allen-Cahn equation in bounded two-dimensional domains. If time allows we also remark on the existence of triple junction solutions defined on the entire plane.

November 24, 2022
Dimitrios Gazoulis (University of Athens, Greece)
On the Γ-convergence of the Allen-Cahn functional with boundary conditions
Abstract. We will briefly introduce the notion of the Γ-convergence, its applications to scalar phase transition type problems and how this is related to the theory of minimal surfaces. Then, we will describe our setting, that is, the study of the ε-energy functional for the vector Allen-Cahn equations and establish the Γ-limit of the functional with boundary conditions. Finally, we will determine some properties of the minimizers of the limiting functional.

November 10, 2022
Grigorios Fournodavlos (University of Crete, Greece)
On the nature of the Big Bang singularity
Abstract. 100 years ago, Friedmann and Kasner discovered the first exact cosmological solutions to Einstein’s equations, revealing the presence of a striking new phenomenon, namely, the Big Bang singularity. Since then, it has been the object of study in a great deal of research on general relativity. However, the nature of the ‘generic’ Big Bang singularity remains a mystery. Rivaling scenarios are abound (monotonicity, chaos, spikes) that make the classification of all solutions a very intricate problem. I will give a historic overview of the subject and describe recent progress that confirms a small part of the conjectural picture.

October 20, 2022
Manousos Maridakis (Aristotle University of Thessaloniki, Greece)
The Concentration Principle for Dirac operators
Abstract. We study the spectrum and the properties of perturbed Dirac operators of the form D_s = D + sA: Γ(E)→Γ(F) on compact Riemannian manifolds (X,g), with vector bundles Ε ⊕ F → X and symbol c and perturbated terms, linear operators A: E→F with s >> 0. A simple relation between c and A, guarantees that the spectrum of the perturbated operator D_s is distinguished into slow and fast eigenvalues as s→∞. The eigenfunctions of slow eigenvalues are concentrated as s→∞ around the singular set Z_A⊂ X of A. We illustrate many interesting examples.

October 6, 2022
Vasiliki Bitsouni (University of Athens, Greece)
On the quasi-steady-state assumption in enzyme kinetics: rigorous analysis
Abstract. We present, from a purely quantitative point of view, the quasi-steady-state assumption for the fundamental mathematical model of the general enzymatic reaction. In particular, we define the two parts of the assumption in a quantitative fashion, we employ a simple algorithm for the proper scaling of the corresponding problem which naturally provides us with the necessary and sufficient information, and we comment, among other issues, on a dispensable third part of the assumption. Joint work with Nikolaos Gialelis (National and Kapodistrian University of Athens) and Ioannis G. Stratis (National and Kapodistrian University of Athens).

September 29, 2022
Phuoc-Tai Nguyen (Masaryk University, Czech Republic)
Nonlinear Schrödinger equations with a singular potential: global existence versus finite time blowup
Abstract. In this talk, I will discuss the focusing nonlinear Schrödinger (NLS) equation with an inverse-square potential. The presence of the potential yields substantial difficulties and requires a fine analysis. I will show the existence ofa minimizer of Hardy-Gagliardo-Nirenberg inequality and the uniqueness of the ground state solution to the NLS equation. As a consequence, I will show that any minimizer can be represented by the ground state solution. Then I will establish global existence versus finite time blowup for the focusing NLS equation.

September 23, 2022
Matteo Rizzi (University of Giessen, Germany)
Symmetry results for the Cahn-Hilliard equation
Abstract. In the talk we will give a classification of the entire solutions to the Cahn-Hilliard equation according to their symmetry properties, such as radial and cylindrical symmetry. We will see that this classification is related to the theory of embedded constant mean curvature surfaces.