### WADE: from Lisbon to the world

The * Lisbon Webinar in Analysis in Differential Equations *is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.

All seminars will take place in the Zoom platform. The links for the planned seminars can be found here. In order to get the password to access the seminars, please subscribe the announcements in the registration menu, or contact one of the organizers.

**Next webinar** **| https://wade.ulisboa.pt/**

To be held online at: https://videoconf-colibri.zoom.us/j/86873402503

Online access password: lisbonwade

**18.11.2021 | 14:00 — 15:00 ****Europe/Lisbon**

**Giovanni Bellettini, **ICTP and Università di Siena

**On a conjecture of De Giorgi on the first variation of the Modica-Mortola functional**

We shall discuss some (not so recent) results on a 1991 conjecture of De Giorgi concerning the Gamma limit of the square norm of the first variation of the Modica-Mortola functionals, and its relation with the Willmore functional.

**04.11.2021 | 14:00 — 15:00 Europe/Lisbon**

**Kelei Wang**, Wuhan University

**Regularity of transition layers in Allen-Cahn equation **

In this talk I will survey the regularity theory for transition layers in singularly perturbed Allen-Cahn equation, from zeroth order regularity to second order one. Some applications of this regularity theory will also be discussed, including De Giorgi conjecture, classification of finite Morse index solutions and construction of minimal hypersurfaces by Allen-Cahn approximation.

**21.10.2021 | 14:00 — 15:00 Europe/Lisbon**

**Simone Dovetta, **Università degli Studi di Roma "La Sapienza"

#### Action versus energy ground states in nonlinear Schrödinger equations

The talk investigates the relations between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold.

First, we show that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre—Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We see that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers, and we exhibit examples of domains where our results apply.

The matter of the talk refers to joint works with Enrico Serra and Paolo Tilli.

**14.10.2021 | ****16:00 — 17:00**** Europe/Lisbon**

**Juraj Földes, **University of Virginia

** Stochastic approach to boundary regularity of hypoelliptic PDEs**

We will discuss the almost sure behavior of solutions of stochastic differential equations(SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hormander condition. We will introduce large deviations to provide some details of proofs. Furthermore, we apply the stochastic results to the problem of identifying regular points for hypoelliptic diffusions and obtain criteria for well posedness of degenerate equations.

This is a joint work with David Herzog and Marco Carfagnini.

**27.05.2021 | 14:00 — 15:00 ****Europe/Lisbon**

**Julio D. Rossi **(Universidad de Buenos Aires)

** Non Linear Mean Value Properties for Monge-Ampère Equations**

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game.

Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Ampère equation.

Joint work with P. Blanc (Jyvaskyla), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).

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**20.05.2021 | 15:00 — 16:00 Europe/Lisbon**

**Mariana Smit Vega Garcia **(Western Washington University)

** Regularity of almost minimizers with free boundary**

We study almost minimizer for functionals that yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem that explicitly takes noise into account. In this talk, we discuss the regularity of almost minimizers to energy functionals with variable coefficients. This is joint work with Guy David, Max Engelstein & Tatiana Toro.

** Please note that the seminar is one hour later than the usual time **

**13.05.2021 | 14:00 — 15:00 Europe/Lisbon**

**Simone Di Marino **(Università di Genova)

** An entropy minimization approach to second-order variational mean field games**

The use of the entropic regularization and the relative Sinkhorn algorithm in order to calculate optimal transport problems is nowadays widely used.

We propose a new viewpoint on application of this regularization technique to variational mean-field games with quadratic Hamiltonian. In particular we focus on the fact that this regularization amounts to solving second order variational mean-field games, dropping in particular the requirement for the regularization parameter to be small, whenever we are interested in the case with diffusion.

Based on a joint work with J.-D. Benamou, G. Carlier and L. Nenna.

**06.05.2021 | 14:00 — 15:00 Europe/Lisbon**

**Didier Pilod **(University of Bergen)

**Global well-posedness and scattering for the Dysthe equation in **

The Dysthe equation is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. After reviewing the derivation of the Dysthe and related equations, we will focus on the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space . This result is sharp in view of the fact that the flow map cannot be continuous below .

Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space and its dual of square bounded variation functions.

We also prove that the initial-value problem is locally well-posed in ,

Our results extend to the finite depth version of the Dysthe equation.

This talk is based on a joint work with Razvan Mosincat (University of Bergen) and Jean-Claude Saut (Université Paris-Saclay).

**29.04.2021 **

** 14:00 — 15:00 **Europe/Lisbon

**Jan Mass **(IST Austria)

**Homogenisation of discrete dynamical optimal transport**

Many stochastic systems can be viewed as gradient flow ('steepest descent') in the space of probability measures, where the driving functional is a relative entropy and the relevant geometry is described by a dynamical optimal transport problem. In this talk we focus on these optimal transport problems and describe recent work on the limit passage from discrete to continuous.

Surprisingly, it turns out that discrete transport metrics may fail to converge to the expected limit, even when the associated gradient flows converge. We will illustrate this phenomenon in examples and present a recent homogenisation result.

This talk is based on joint work with Peter Gladbach, Eva Kopfer, and Lorenzo Portinale.

******

**15:00 — 16:00**** **Europe/Lisbon

**Max Fathi **(Université de Paris)

**A proof of the Caffarelli contraction theorem via entropic interpolation**

The Caffarelli contraction theorem states that optimal transport maps (for the quadratic cost) from a Gaussian measure onto measures that satisfy certain convexity properties are globally Lipschitz, with a dimension-free estimate. It has found many applications in probability, such as concentration and functional inequalities. In this talk, I will present an alternative proof, using entropic interpolation and variational arguments. Joint work with Nathael Gozlan and Maxime Prod'homme.

**22.04.2021 | 14:00 — 15:00 Europe/Lisbon**

**Juan Luis Vázquez **(Universidad Autónoma de Madrid)

** The theory of fractional p-Laplacian equations**

We consider the time-dependent fractional $p$-Laplacian equation with parameter $p>1$ and fractional exponent $0<$ $s<1$. It is the gradient flow corresponding to the Gagliardo fractional energy. Our main result is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on delicate analysis. We will concentrate on the sublinear or “fast” regime, $1<$ $p<2$, since it offers a richer theory. Fine bounds in the form of global Harnack inequalities are obtained as well as solutions having strong point singularities (Very Singular Solutions) that exist for a very special parameter interval. They are related to fractional elliptic problems of nonlinear eigenvalue form. Extinction phenomena are discussed.

**15.04.2021 **

** 14:00 — 15:00 **Europe/Lisbon

**Nicola Fusco **(Università di Napoli "Federico II")

** Asymptotic stability for the gradient flow of some nonlocal energies**

I will start by discussing some recent results on the asymptotic stability of the $H^{-1}$-gradient flow of the perimeter, the so called surface diffusion. Then I will consider the $H^{-1}$-gradient flow of some energy functionals given by the area of an interface plus a non local volume term.

This is a joint work with E. Acerbi, V. Julin and M. Morini

******

**15:00 — 16:00**** **Europe/Lisbon

**Riccardo Scala **(Università degli Studi di Siena)

** Nonlocality features of the area functional and the Plateau problem**

We briefly discuss the definition of relaxation of the area functional. The relaxed area functional, denoted by $A$, extends the classical area functional, which, for any "regular" map $v:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^N$ evaluates the $n$-dimensional area of its graph over $U$. The problem of determining the domain and the expression of $A$ is open in codimension greater than 1. Specifically, this relaxed functional turns out to be nonlocal and cannot be expressed by an integral formula. We discuss how it is related to classical and nonclassical versions of the Plateau problem. As a main example, we try to understand what is the relaxed graph of the function $x/|x|$, a question that surprisingly remained open for decades.

**Next webinar** **| https://wade.ulisboa.pt/**

To be held online at: https://videoconf-colibri.zoom.us/j/85036762376

Online access password: lisbonwade

**08.04.2021 | 14:00 — 15:00 Europe/Lisbon**

**Valeria Chiadò Piat **(Politecnico di Torino)

**An extension theorem from connected sets and homogenization of non-local functionals**

Extensions operators are a classical tool to provide uniform estimates and gain compactness in the homogenization of integral functionals over perforated domains. In this talk we discuss the case of non-local functionals. The results are obtained in collaboration with Andrea Braides and Lorenza D'Elia.

**01.04.2021 | 14:00 — 15:00 Europe/Lisbon**

**Grégoire Allaire **(CMAP, École Polytechnique)

**Some problems and some solutions in shape and topology optimization of structures built by additive manufacturing**

Additive manufacturing (or 3-d printing) is a new exciting way of building structures without any restriction on their topologies. However, it comes with its own difficulties or new issues. Therefore, it is a source of many interesting new problems for optimization. I shall discuss two of them and propose solutions to these problems, but there is still a lot of room for improvement!

First, additive manufacturing technologies are able to build finely graded microstructures (called lattice materials). Their optimization is therefore an important issue but also an opportunity for the resurrection of the homogenization method ! Indeed, homogenization is the right technique to deal with microstructured materials where anisotropy plays a key role, a feature which is absent from more popular methods, like SIMP. I will describe recent work on the topology optimization of these lattice materials, based on a combination of homogenization theory and geometrical methods for the overall deformation of the lattice grid.

Second, additive manufacturing, especially in its powder bed fusion technique, is a very slow process because a laser beam must travel along a trajectory, which covers the entire structure, to melt the powder. Therefore, the optimization of the laser path is an important issue. Not only do we propose an optimization strategy for the laser path, but we couple it with the usual shape and topology optimization of the structure. Numerical results show that these two optimizations are tightly coupled.

This is a joint work with many colleagues, including two former PhD students, P. Geoffroy-Donders and M. Boissier.

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* ***25.03.2021 | 11:00 — 12:00 Europe/Lisbon**

**Yukihiko Nakata **(Aoyama Gakuin University, Tokyo)

**Period two solution for a class of distributed delay differential equations**

We consider a periodic solution for a class of distributed delay differential equations.

A period two solution for distributed delay differential equations, where the period is twice the maximum delay, is shown to satisfy a Hamiltonian system of ordinary differential equations, from which we can construct the period two solution for the distributed delay differential equation.

The idea is based on Kaplan & Yorke (1974, JMAA) for a discrete delay differential equation. We present distributed delay differential equations that have periodic solutions expressed in terms of the Jacobi elliptic functions.

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**18.03.2021 | 14:00 — 15:00 Europe/Lisbon**

**Gabriele Benomio** (Princeton University)

**A new instability for higher dimensional black holes**

The dynamics of solutions to the Einstein equations is richer in dimensions higher than . In contrast with the classical stability of stationary, asymptotically flat black hole solutions in dimensions, some families of higher dimensional black holes suffer from dynamical instabilities. I will discuss a subtle instability affecting a wide class of higher dimensional black holes which has not been previously observed in the literature. This new instability is, in a sense, more fundamental than the other known instability phenomena in higher dimensions and can be related to a precise geometric property of the class of spacetimes considered.

******

**18.03.2021 | 15:00 — 16:00 Europe/Lisbon**

**Shrish Parmeshwar **(University of Bath)

**Global-in-Time Solutions to the N-Body Euler-Poisson System**

** **We investigate the -Body compressible Euler-Poisson system, modelling multiple stars interacting with each other via Newtonian gravity. If we prescribe initial data so that each star expands indefinitely, one might expect that two of them will collide in finite time due to their expansion, and the influence of gravity. In this talk we show that there exists a large family of initial positions and velocities for the system such that each star can expand for all time, but no two will touch in finite time. To do this we use scaling mechanisms present in the compressible Euler system, and a careful analysis of how the gravitational interaction between stars affects their dynamics.

**11.03.2021 | 14:00 — 15:00 Europe/Lisbon**

**Bruno Premoselli **(Université Libre de Bruxelles)

**Towers of bubbles for Yamabe-type equations in dimensions larger than 7**

In this talk we consider perturbations of Yamabe-type equations on closed Riemannian manifolds. In dimensions larger than 7 and on locally conformally flat manifolds we construct blowing-up solutions that behave like towers of bubbles concentrating at a critical point of the mass function. Our result does not assume any symmetry on the underlying manifold.

We perform our construction by combining finite-dimensional reduction methods with a linear blow-up analysis in order to sharply control the remainder of the construction in strong spaces. Our approach works both in the positive and sign-changing case. As an application we prove the existence, on a generic bounded open set of $\mathbb{R}^n$, of blowing-up solutions of the Brézis-Nirenberg equation that behave like towers of bubbles of alternating signs.

**04.03.2021 | 14:00 — 15:00 Europe/Lisbon**

**Bozhidar Velichkov **(Università di Pisa)

**Vectorial free boundary problems**

The vectorial Bernoulli problem is a variational free boundary problem involving the Dirichlet energy of a vector-valued function and the measure of its support. It is the vectorial counterpart of the classical one-phase Bernoulli problem, which was first studied by Alt and Caffarelli in 1981.

In this talk, we will discuss some results on the regularity of the vectorial free boundaries obtained in the last years by Caffarelli-Shahgholian-Yeressian, Kriventsov-Lin, Mazzoleni-Terracini-V., and Spolaor-V.. Finally, we will present some new results on the rectifiability of the singular set obtained in collaboration with Guido De Philippis, Max Engelstein and Luca Spolaor.

***

**04.03.2021 | 15:00 — 16:00 Europe/Lisbon**

**Dario Mazzoleni **(Università di Pavia)

**Regularity of the optimal sets for the second Dirichlet eigenvalue**

** **First of all, we recall the basic notions and results concerning shape optimization problems for the eigenvalues of the Dirichlet Laplacian.

Then we focus on the study of the regularity of the optimal shapes and on the link with the regularity of related free boundary problems.

The main topic of the talk is the regularity of the optimal sets for a "degenerate'" functional, namely the second Dirichlet eigenvalue in a box. Given an open and bounded set of class we consider the following shape optimization problem, for ,

where denotes the second eigenvalue of the Dirichlet Laplacian on .

In this talk we show that any optimal set for (1) is equivalent to the union of two disjoint open sets, , which are regular up to a (possibly empty) closed singular set of Hausdorff dimension at most , which is contained in the one-phase free boundaries.

In particular, we are able to prove that the set of two-phase points, that is, , is contained in the regular set.

This is a joint work with Baptiste Trey and Bozhidar Velichkov.

** **

To be held online at: https://videoconf-colibri.zoom.us/j/81421184299

Online access password: lisbonwade

**25.02.2021 | 14:00 — 15:00 Europe/Lisbon**

**Boyan Sirakov **(PUC - Rio)

**The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients**

In this joint work with P. Souplet we develop a new, unified approach to the following two classical questions on elliptic PDE:

(i) the strong maximum principle for equations with non-Lipschitz nonlinearities; and

(ii) the at most exponential decay of solutions in the whole space or exterior domains.

Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

**18.02.2021 | Thursday | 14:00 — 15:00 Europe/Lisbon**

**Paolo Gidoni **(Czech Academy of Sciences)

**A vanishing inertia analysis for finite dimensional rate-independent systems and an application to soft crawlers**

The quasistatic limit is a convenient approximation in the modelling of several (suitable) mechanical systems, when the evolution occurs at a sufficiently slow time-scale. In this talk we discuss the validity of the quasistatic approximation in finite-dimensional rate-independent systems via a vanishing-inertia asymptotic analysis of dynamic evolutions. More precisely, we show the uniform convergence of dynamic solutions to the quasistatic one, employing the concept of energetic solution. Our work is motivated by the application to a family of models for biological and bio-inspired crawling locomotion. Hence a part of the seminar will focus on modelling: we will discuss how soft crawlers can be effectively described in our theoretical framework and briefly survey the relevance, or lack thereof, of inertia in some locomotion strategies. By a technical point of view, our application requires time-dependence of the dissipation potential and translation invariance of the potential energy.

**11.02.2021 | Thursday | 14:00 — 15:00 Europe/Lisbon**

**Harbir Antil **(George Mason University)

**Fractional PDEs: Control, Numerics, and Applications**

Fractional calculus and its application to anomalous diffusion has recently received a tremendous amount of attention. In complex/heterogeneous material mediums, the long-range correlations or hereditary material properties are presumed to be the cause of such anomalous behavior. Owing to the revival of fractional calculus, these effects are now conveniently modeled by fractional-order differential operators and the governing equations are reformulated accordingly. Similarly, the potential of fractional operators has been harnessed in various scientific domains like geophysical electromagnetics, imaging science, deep learning, etc.

In this talk, fractional operators will be introduced and both linear and nonlinear, fractional-order differential equations will be discussed. New notions of optimal control and optimization under uncertainty will be presented. Several applications from geophysics, imaging science, and deep learning will be presented.

**04.02.2021 | Thursday | 14:00 — 15:00 Europe/Lisbon**

**Yvan Martel **(École Polytechnique)

** On the asymptotic stability of kinks for (1+1)-scalar field models**

The talk concerns stability properties of kinks for (1+1)-dimensional nonlinear scalar field models of the form

We establish a simple and explicit sufficient condition on the potential for the asymptotic stability of a given moving or standing kink.

We present applications of the criterion to some models of the Physics literature.

Work in collaboration with Michał Kowalczyk, Claudio Muñoz and Hanne Van Den Bosch. See also the related work with Michał Kowalczyk and Claudio Muñoz.

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**28.01.2021 | ****Thursday**** | 14:00 — 15:00 Europe/Lisbon**

**Enrico Serra **(Politecnico di Torino)

** NLS ground states on metric trees: existence results and open questions**

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Polya-Szego inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder.

**21.01.2021 | 14:00 — 15:00 Europe/Lisbon**

**Charles Collot, **(Cergy Paris Université)

**On the stability of equilibria for infinitely many particles**

We study the evolution of a system of particles. Instead of the usual Hartree equation for density matrices, we consider the following equivalent model, proposed by de Suzzoni, of a Hartree type equation but for a random field:

Above, is a time-dependent random field, a pair interaction potential, *the convolution product and the expectation. This equation admits equilibria which are random Gaussian fields whose laws are invariant by time and space translations. They are hence not localised and represent an infinite number of particles. We give a stability result under certain hypotheses, by showing that small perturbations scatter as to linear waves. This is joint work with de Suzzoni.

**14.01.2021 | 10:00 — 12:00 Europe/Lisbon**

**Serena Dipierro & Enrico Valdinoci **(University of Western Australia)

**Nonlocal Minimal Surfaces: interior regularity, boundary behavior and stickiness phenomena**

Surfaces which minimize a nonlocal perimeter functional exhibit quite different behaviors than the ones minimizing the classical perimeter. We will investigate some structural properties of nonlocal minimal surfaces both in the interior of a given domain and in the vicinity of its boundary. Among these peculiar features, an interesting property, which is also in contrast with the pattern produced by the solutions of linear equations, is given by the capacity, and the strong tendency, of adhering at the boundary. We will also discuss this phenomenon and present some recent results.

(These are two consecutive talks: Part I is given by Serena Dipierro, Part II by Enrico Valdinoci)

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**07.01.2021 | 16:30 — 17:30 Europe/Lisbon**

**Nilima Nigam **(Simon Fraser University)

**Boundary integral strategies for the Steklov eigenproblem**

In Steklov eigenproblems for elliptic operators, the spectral parameter links boundary traces of eigenfunctions to traces of the Neumann data. It is natural, therefore, to reformulate such eigenproblems in terms of boundary integral operators, which allow for nonsmooth boundaries. In this talk we describe such strategies in the context of Steklov problems for the Laplacian as well as the Helmholtz operator, and their use in studying questions arising in spectral geometry.

**17.12.2020 | 14:30 — 15:30 Europe/Lisbon**

**Svetlana Roudenko **(Florida International University)

** **

**Zakharov-Kuznetsov equation: toward soliton resolution**

We consider Zakharov-Kuznetsov (ZK) equation, which is a higher-dimensional version of the Korteweg-de Vries (KdV) equation, and investigate the dynamics of solutions, especially questions about the soliton stability. We first discuss the situation in two dimensions, in particular, the instability of solitons in the 2d cubic (critical) ZK equation, which leads to blow-up. Then we consider the 3d quadratic ZK equation, originally introduced by Zakharov and Kuznetsov in early 1970's, and discuss the asymptotic stability of solitons. We will also show numerical findings on the formation of solitons and radiation in this equation. This talk will be based on joint works with L.G. Farah, J. Holmer, C. Klein, N. Stoilov, K. Yang.

**10.12.2020 | 14:30 — 15:30 Europe/Lisbon**

**Qiao Huang **(GFM, Universidade de Lisboa)

** **

**Stochastic Cucker-Smale model: collision-avoidance and flocking**

In this talk, we consider the Cucker-Smale flocking model involving both singularity and noise. We first show the local strong well-posedness for the system, in which the communication weight is locally Lipschitz beyond the origin. Then, for the special case that the communication weight has a strong singularity at the origin, we establish the global well-posedness by showing the finite time collision-avoidance. Finally, we study the large time behavior of the system when the communication weight is of zero lower bound. The conditional flocking emerges for the case of constant noise intensity, while the unconditional flocking emerges for various time-varying intensities and long-range communications.

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**03.12.2020 | 14:30 — 15:30 Europe/Lisbon**

**Youcef Mammeri **(Université de Picardie Jules Verne)

** **

**A SIR-type model with diffusion to describe the spatial spread of Covid-19**

We all have to deal with the coronavirus epidemic. Many strategies have been put in place to try to contain the disease, with varying success.

I will present an SIR-type mathematical model to predict the state of the epidemic. The effect of distancing, isolation of exposed individuals and treatment of symptoms will be compared.

I will begin with a simple explanation of SIR models, then discuss a PDE model and its resolution.

**26.11.2020 | 13:30 — 14:30 Europe/Lisbon**

**Jean-Francois Babadjian **(Université Paris Saclay)

** **

**Concentration versus oscillation effects in brittle damage**

This talk is concerned with the asymptotic analysis of a variational model of brittle damage, when the damaged zone concentrates into a set of zero Lebesgue measure, and, at the same time, the stiffness of the damaged material becomes arbitrarily small. In a particular non-trivial regime, concentration leads to a limit energy with linear growth as typically encountered in perfect plasticity. While the singular part of the limit energy can be easily described, the identification of the bulk part of the limit energy requires a subtler analysis of the interplay between concentration and oscillation properties of the displacements.

This is a joint work with F. Iurlano and F. Rindler.

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**26.11.2020 | 14:30 — 15:30 Europe/Lisbon**

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**Marco Caroccia **(Università degli studi di Roma Tor Vergata)

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**Contact surface of Cheeger sets**

Geometrical properties of Cheeger sets have been deeply studied by many authors since their introduction, as a way of bounding from below the first Dirichlet (p)-Laplacian eigenvualue. They represents the first eigenfunction of the Dirichlet (1)-Laplacian of a domain. In this talk we will introduce a recent property, studied in collaboration with Simone Ciani (Università degli studi di Firenze), concerning their contact surface with the ambient space. In particular we will show that the contact surface cannot be too small, with a lower bound on the dimension strictly related to the regularity of the ambient space. The talk will focus on the introduction of the problem. It will start with a brief explanation of its connection with the Dirichlet (p)-Laplacian eigenvalue problem. Then a brief sketch of the proof is given. Functional to the whole argument is the notion of removable singularity, as a tool for extending solutions of pdes under some regularity constraint.

**26.11.2020 | 13:30 — 14:30 Europe/Lisbon**

**Jean-Francois Babadjian **(Université Paris Saclay)

** **

**Concentration versus oscillation effects in brittle damage**

This talk is concerned with the asymptotic analysis of a variational model of brittle damage, when the damaged zone concentrates into a set of zero Lebesgue measure, and, at the same time, the stiffness of the damaged material becomes arbitrarily small. In a particular non-trivial regime, concentration leads to a limit energy with linear growth as typically encountered in perfect plasticity. While the singular part of the limit energy can be easily described, the identification of the bulk part of the limit energy requires a subtler analysis of the interplay between concentration and oscillation properties of the displacements.

This is a joint work with F. Iurlano and F. Rindler.

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**26.11.2020 | 14:30 — 15:30 Europe/Lisbon**

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**Marco Caroccia **(Università degli studi di Roma Tor Vergata)

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**Contact surface of Cheeger sets**

Geometrical properties of Cheeger sets have been deeply studied by many authors since their introduction, as a way of bounding from below the first Dirichlet (p)-Laplacian eigenvualue. They represents the first eigenfunction of the Dirichlet (1)-Laplacian of a domain. In this talk we will introduce a recent property, studied in collaboration with Simone Ciani (Università degli studi di Firenze), concerning their contact surface with the ambient space. In particular we will show that the contact surface cannot be too small, with a lower bound on the dimension strictly related to the regularity of the ambient space. The talk will focus on the introduction of the problem. It will start with a brief explanation of its connection with the Dirichlet (p)-Laplacian eigenvalue problem. Then a brief sketch of the proof is given. Functional to the whole argument is the notion of removable singularity, as a tool for extending solutions of pdes under some regularity constraint.

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**16.11.2020**** | 14:30 — 15:30 Europe/Lisbon**

**Hynek Kovařík **(Università degli studi di Brescia)

** **

**Absence of positive eigenvalues of magnetic Schroedinger operators**

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schroedinger operators in . In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller-Simon shows that our result is sharp as far as the decay of the magnetic field is concerned.

The talk is based on a joint work with Silvana Avramska-Lukarska and Dirk Hundertmark.

**Please note that this seminar will take place Monday, not the usual day of the Lisbon WADE.**

**12.11.2020 | 14:30 — 15:30 Europe/Lisbon**

**Rainer Mandel **(Karlsruhe Institute of Technology)

**New dual variational methods for Nonlinear Helmholtz Equations and polychromatic solutions of Nonlinear wave equations**

In the first part of my talk, I present the classical dual variational method in the context of Nonlinear Helmholtz equations that describe monochromatic waves in nonlinear materials. Afterwards, I discuss two recent generalizations of the method. The first deals with an extension to Nonlinear Helmholtz equations with sign-changing nonlinearities. For these problems we construct solutions that have infinite Morse-Index in the dual variational formulation. The second generalization concerns dual variational methods for the construction of breathers, i.e., polychromatic, spatially localized and time-periodic solutions of nonlinear wave equations.

The results were obtained in collaboration with D. Scheider and T. Yesil.

**5/11/2020, 14:30 — 15:30 Europe/Lisbon**

**Wladimir Neves **(Universidade Federal do Rio de Janeiro)** **

**Homogenization of Schrödinger equations. Extended Effective Mass Theorems for non-crystalline matter.**

In this talk, we study the homogenization of the Schrödinger equation beyond the periodic setting. More precisely, rigorous derivation of the effective mass theorems in solid state physics for non-crystalline materials are obtained.

This is a joint work with Vernny Ccajma and Jean Silva.

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**29/10/2020, 13:30 — 14:30 Europe/Lisbon**

**Lorenzo Mazzieri and Piotr Chruściel **(Università di Trento / University of Vienna)

**Static vacuum black holes and Lambda**

We will review the status of the uniqueness theory of static vacuum black holes, with or without a cosmological constant and we will outline the proof of a uniqueness theorem with , proved jointly in collaboration with Stefano Borghini.

**22/10/2020, 14:30 — 15:30 Europe/Lisbon**

**Jean-Baptiste Casteras **(CMAFcIO, Universidade de Lisboa)

**Standing wave and travelling wave solutions for a fourth order Schrödinger equation**

In this talk, we will be interested in standing wave solutions to a fourth order nonlinear Schrödinger equation having second and fourth order dispersion terms. This kind of equation naturally appears in nonlinear optics. In a first time, we will establish the existence of ground-state and renormalized solutions. We will then be interested in their qualitative properties, in particular their stability.

Joint works with Denis Bonheure, Ederson Moreira Dos Santos, Tianxiang Gou, Louis Jeanjean and Robson Nascimento.

**04/09/2020, ****11:00–12:00**** Europe/Lisbon — Online ***

**Marco Morandotti **(Politecnico di Torino)

**Spatially inhomogeneous evolutionary games**

We study an interaction model of a large population of players based on an evolutionary game, which describes the dynamical process of how the distribution of strategies changes in time according to their individual success.

We assume that the population of players is distributed over a state space and that they are each endowed with probability distributions of pure strategies, which they draw at random to evolve their states. Simultaneously, the mixed strategies evolve according to a replicator dynamics, modeling the success of pure strategies according to a payoff functional.

We establish existence, uniqueness, and stability of Lagrangian and Eulerian solutions of this dynamical game by using methods of ODE and optimal transport on Banach spaces.

We apply the general theoretical framework to perform the mean-field analysis of a multi-population agent-based model, where a particle dynamics derived by a nonlocal velocity and a Markov-type jump process for the label s are coupled.

An alternate Lagrangian approximation scheme at discrete times is also proposed.

This research is from joint works with S. Almi, L. Ambrosio, M. Fornasier, G. Savaré, and F. Solombrino.

**14/07/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Pavel Exner **(Doppler Institute for Mathematical Physics and Applied Mathematics, Prague)

**Vertex coupling and spectra of periodic quantum graphs.**

The talk focuses on the influence of the vertex coupling on spectral properties of periodic quantum graphs. Specifically, two questions will be addressed. The first concerns the number of open spectral gaps; it is shown that graphs with a nontrivial coupling can have finite but nonzero number of them. Secondly, motivated by recent attempts to model the anomalous Hall effect, we investigate a class of vertex couplings that violate the time reversal invariance. For the simplest coupling of this type we show that its high-energy properties depend on the parity of the lattice vertices, and discuss various consequences of this property.

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**09/07/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Elvira Zappale **(Università Degli Studi di Salerno)

**Optimal design problems**I will present several integral representation results for certain functionals arising in the context of optimal design and damage models, in presence of a perimeter penalization term.

I will consider several frameworks, and I will also discuss the case with non-standard growth conditions.

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**30/06/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Barbara Brandolini **(Università Degli Studi di Napoli Federico II)

**Sharp lower bounds for Neumann eigenvalues**

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**23/06/2020, Tuesday, 11:00–12:00 Europe/Lisbon — Online ***

**Tatsuya Miura **(Tokyo Institute of Technology)

**On the isoperimetric inequality and surface diffusion flow for multiply winding curves.**

In this talk we discuss dynamical stability of multiply covered circles under the surface diffusion flow. To this end we first establish a general form of the isoperimetric inequality for immersed closed curves under rotational symmetry, which would be of independent interest. We then apply it to obtaining a certain class of rotationally symmetric initial curves from which solutions to the surface diffusion flow exist globally-in-time and converge to multiply covered circles. This talk is based on a joint work with Shinya Okabe at Tohoku University.

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**16/06/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Riccardo Adami **(Politecnico di Torino)

**Ground states of the Nonlinear Schroedinger Equation on Graphs: an overview.**

Driven by physical and technological applications, during the last five years the study of nonlinear evolution on branched structures (graphs, networks) has undergone a fast development. We review on the main achievements and on the open problems. This is a joint project with several people, among which Simone Dovetta, Enrico Serra, Lorenzo Tentarelli, and Paolo Tilli.

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**09/06/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Lucio Boccardo **(Università di Roma La Sapienza)

**Recent developments on Dirichlet problems with singular convection/drift terms.**

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* *

**02/06/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Maria Colombo **(École Polytechnique Fédérale de Lausanne)

**Nonunique characteristic curves of Sobolev vector fields**

Given a vector field in , the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth; this, in turn, translates in existence and uniqueness results for the transport equation. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollaryof the uniqueness of the trajectory of the ODE for a.e. initial datum. In this talk we give an overview of the topic and we provide a negative answer to this question. To show this result we exploit the connection with the transport equation, based on Ambrosio’s superposition principle, and a new ill-posedness result for positive solutions of the continuity equation.

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**26/05/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Luis Vega** (Basque Center for Applied Mathematics)

**The Vortex Filament Equation, the Talbot effect, and non-circular jets**

We will propose the vortex filament equation as a possible toy model for turbulence, in particular because of its striking similarity to the dynamics of non-circular jets. This similarity implies the existence of some type of Talbot effect due to the interaction of non-linear waves that propagate along the filament. Another consequence of this interaction is the existence of a new class of multi-fractal sets that can be seen as a generalization of the graph of Riemann’s non-differentiable function. Theoretical and numerical arguments about the transfer of energy will be also given. This a joint work with V. Banica and F. de la Hoz.

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**19/05/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Bernold Fiedler** (Institute of Mathematics, Freie Universität Berlin)

**Sturm meanders: global attractors, Temperley-Lieb algebras, and black holes**

Fusco and Rocha studied Neumann boundary value problems for scalar ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutations. These permutations relate, on the one hand, to a special class of meandering curves as introduced by Arnol’d in a singularity theory context. On the other hand, they became central in the study of global attractors of nonlinear parabolic partial differential equations of Sturm type.

We discuss relations of Sturm meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.

We report on joint work with Pablo Castañeda, Juliette Hell, Carlos Rocha, and Brian Smith. See also http://dynamics.mi.fu-berlin.de/

For further material we recommend the beautifully illustrated book “Meanders” by Anna Karnauhova, de Gruyter 2017.

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**12/05/2020, Tuesday, 16:30–17:30 Europe/Lisbon — Online ***

**Diogo Oliveira e Silva **(University of Birmingham)

**Global maximizers for spherical restriction**

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**05/05/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online ***

**Filippo Santambrogio** (Université Claude Bernard - Lyon 1)

**Optimal transport methods for the regularity of 2D functions of least gradient**

The least gradient problem (minimizing the BV norm with given boundary data), motivated by both image processing applications and connections with minimal surfaces, is known to be equivalent, in the plane, to the Beckmann minimal-flow problem (an alternative formulation of the *L*^{1} Monge-Kantorovich optimal transport problem) with source and target measures located on the boundary of the domain. Hence, Sobolev regularity of functions of least gradient is equivalent in this setting to *L ^{p} *bounds on the solution of the Beckmann problem (i.e. on what is called the transport density) and can be attacked with techniques which are now standard in optimal transport. From the transport point of view, the novelty of the estimates that I will present, coming from a joint paper with S. Dweik, lies in the fact they are obtained for transport between measures which are concentrated on the boundary. From the BV point of view, a new result is the

*W*

^{1,p}^{ }regularity of the least gradient function whenever the boundary datum is

*W*

^{1,p}^{ }as a 1

*D*function: moreover, the optimal transport framework is strong enough to deal with arbitrary strictly convex norms instead of the Euclidean one with almost no effort.

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* *

*The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.*

**28/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online**

**Michael Goldman** (Laboratoire Jacques-Louis Lions and Université Paris 7)

**On an old conjecture of Almgren**

In this talk I will give an overview on the few results available on the conjecture of Almgren regarding the convexity of drops subject to the action of an external potential. In particular I will present recent progress in this direction obtained with G. De Philippis on their connectedness. Together with an older result of McCann, this answers positively the conjecture in dimension two. The proof is inspired by the two-point function technique introduced by B. Andrews and is reminiscent of the doubling of variables trick in the context of viscosity solutions.

**21/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online**

**Matthias Röger**, Technische Universität Dortmund

**An obstacle type problem as a limit of a model for cell polarization**

We consider the polarization of a cell in response to an outer signal. The mathematical model consists of a diffusion equation in the inner volume coupled to a reaction diffusion system on the cell membrane. In a certain asymptotic limit we rigorously prove the convergence towards a generalized obstacle problem. In term of this limit system we derive conditions for the onset of polarization. The results will be mainly presented for the stationary case, we will briefly discuss some extensions to the time-dependent case.

(This is joint work with Barbara Niethammer, Juan Velazquez, and Anna Logioti)

**14/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online**

**Dorin Bucur,** Université de Savoie.

**Boundary behaviour of Robin problems and isoperimetric spectral inequalities.**

Consider the Poisson equation with Robin boundary conditions in a (nonsmooth) domain with a bounded, nonnegative right hand side. Given a point on the boundary, the question is whether the solution has a strictly positive lower limit at this point. If the domain is smooth the answer is positive as a consequence of the Hopf maximum principle. If the domain is not smooth, the answer may be positive or negative, depending on the geometry of the domain around the point. This question was raised in a probabilistic context by Bass, Burdzy and Chen in 2008, when they obtained results for Lipschitz sets and cuspidal domains.

Our motivation is related to the fact that positive answers to the question above, together with a control of the infimum of the boundary values, lead to sharp quantitative forms of isoperimetric inequalities of spectral type for the Robin Laplacian.

In this talk, I will make the point on recent advances on isoperimetric inequalities involving the Robin Laplacian and I will show how the boundary behaviour pops up in the quantitative form of the inequalities. If times remains, I will present a variational approach to deal with the boundary behaviour for general elliptic operators and various geometric contexts.

The results were obtained together with A. Giacomini and M. Nahon.

**07/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online**

Susanna Terracini, Università di Torino.

Pattern Formation Through Spatial Segregation.

*Video available here.*

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