# Lisbon WADE — Webinar in Analysis and Differential Equations

## Past sessions

### 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.

### 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.

### 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.

**Note that the link is not the usual one.**

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

We investigate the $N$-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.

### A new instability for higher dimensional black holes

The dynamics of solutions to the Einstein equations is richer in dimensions higher than $3+1$. In contrast with the classical stability of stationary, asymptotically flat black hole solutions in $3+1$ 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.

### 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.

### 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 $D\subset \mathbb{R}^d$ an open and bounded set of class $C^{1,1}$, we consider the following shape optimization problem, for $\Lambda>0$,$$\label{eq:main}\min{\Big\{\lambda_2(A)+\Lambda |A| : A\subset D,\text{ open}\Big\}},$$where $\lambda_2(A)$ denotes the second eigenvalue of the Dirichlet Laplacian on $A$.

In this talk we show that any optimal set $\Omega$ for \eqref{eq:main} is equivalent to the union of two disjoint open sets, $\Omega^\pm$, which are $C^{1,\alpha}$ regular up to a (possibly empty) closed singular set of Hausdorff dimension at most $d-5$, 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, $\partial \Omega^+\cap \partial \Omega^-\cap D$, is contained in the regular set.

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

### 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.

### 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.

### 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.

### 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.

### 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 $\partial_t^2 \phi - \partial_x^2 \phi + W'(\phi) = 0 \quad (t,x) \in {\bf \mathbb R}\times {\mathbb R}.$ We establish a simple and explicit sufficient condition on the potential $W$ 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.

### 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.

### 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:$$iX_t=-\Delta X +(w*\mathbb E(|X|^2))X.$$Above, $X:[0,T]\times \mathbb R^d\times \Omega$ is a time-dependent random field, $w$ a pair interaction potential, $*$ the convolution product and $\mathbb E$ 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 $t\rightarrow \pm \infty$ to linear waves. This is joint work with de Suzzoni.

### 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)

Please note that the seminar takes place at 10am Lisbon/London time, not the usual time for Lisbon WADE

### 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.

Please not that the talk will be at 16:30 Lisbon/London time.

Upcoming talks:

14 Jan - Serena Dipierro & Enrico Valdinoci, University of Western Australia

21 Jan - Charles Collot, Cergy Paris Université

28 Jan - Enrico Serra, Politecnico di Torino

### 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.

Projecto FCT UIDB/04459/2020.

### 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.

### 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.

Projecto FCT UIDB/04459/2020.

### 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.

Older session pages: Previous 2 3 Oldest

All seminars will take place in the Zoom platform which you need to install (although you don't need to register). In order to get the password to access the seminars, please subscribe the announcements or contact the organizers.

Organizers: Hugo Tavares, James Kennedy and Nicolas Van Goethem

Joint iniciative of the research centers CAMGSD, CMAFcIO and GFM.