Titles and abstracts

Invited talks

Short talks

Presentations

Paolo Acampora 

Title: Thin insulating layers: results and techniques

Abstract: We inspect some properties of minimum and minimizers of functionals that represent the physical situation in which a thin layer of insulating material is placed around a fixed body which one could aim to insulate. We discuss the case in which the thermal exchange with the enviroment occurs through convection, that corresponds to Robin boundary conditions.  

Vincenzo Amato

Title: Some asymptotic quantitative inequalities.

Abstract: The classical quantitative isoperimetric inequality has opened the way for a rich line of research into quantitative versions of many inequalities. In practice, such an inequality aims to estimate how much a set with an almost minimal perimeter must be 'similar' to a ball.

We will discuss the case where a spectral inequality is not achieved. In particular, we will analyse the case of two inequalities, one concerning torsional rigidity and the other the first non-trivial Neumann eigenvalue, both of which are asymptotically achieved by a sequence of thinning rectangles.

Michiel van den Berg

Title:  "Maximising the product of torsional rigidity and Newtonian capacity among convex sets with a perimeter constraint"

Abstract: It is shown that the maximum of the product of torsional rigidity and Newtonian capacity among convex sets in R^d (d>2) with fixed perimeter 1 is maximised by the ball with perimeter 1. A key lemma is an isoperimetric upper bound for the Newtonian capacity.  A quantitative refinement of the latter bound is also obtained.

Konstantinos Bessas

TITLE: NON-LOCAL BV FUNCTIONS AND A DENOISING MODEL WITH L^1 FIDELITY

ABSTRACT:

We study a general total variation denoising model with weighted L^1 fidelity,

where the regularizing term is a non-local variation induced by a suitable (non-integrable)

kernel K, and the approximation term is given by the L^1 norm with respect to a non-singular

measure with positively lower-bounded L^∞ density.-

We provide a detailed analysis of the space of non-local BV functions with finite total K-

variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings

and isoperimetric and monotonicity properties of the K-variation and the associated K-

perimeter.

Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the

study of the fidelity in our model.

This is a joint work with Giorgio Stefani (SISSA, Trieste).

Giulia Bevilacqua 

Title: A variational model for nematic soap films. 

Abstract: In this talk I discuss the existence and the derivation of some geometrical properties of minimizers for the energy functional penalizing both the area contribution, as in the classical Plateau problem, and the Frank free-energy, typically arising from soft materials like nematic liquid crystal. This is a joint work with Chiara Lonati, Luca Lussardi and Alfredo Marzocchi

Francesca Bianchi

Title: Geometric estimates for the first eigenvalue of powers of Laplacian

Abstract: In this talk we consider the $s$--powers of the Dirichlet--Lapacian, with $1<s\le 2$, and the class of open sets with finite inradius in dimensions $2$ and $3$.

Our goal is to discuss the existence of lower bounds for the first eigenvalue of such powers of the Dirichlet--Laplacian in terms of the  inradius of the set.

To this aim we exploit the notion of $s$--capacity and some of its properties and we show that such estimates exist only when $s>N/2$. 

Some of the results presented are obtained in collaboration with Lorenzo Brasco (University of Ferrara).

Guy Bouchitté

Title: The optimal grillage problem

Luca Briani

Title: Hexagonal structures and  a mean-to-max problem for the  torsion function.

Abstarct: Hexagonal structures appear frequently in various contexts. I will recall some of them, related to shape optimization problems. I will then present the study of a mean-maximum functional for the torsion function and discuss the emergence of an hexagonal structure in this case.

Lorenzo Brasco

Ttile: An inequality by E.Makai and W. K. Hayman

Abstract: We discuss lower bounds on the sharp Poincaré constant for planar open sets, in terms of their inradius. We recall some classical results and discuss some open problems and generalizations. 

We will present some results obtained in collaboration with Francesca Bianchi (Parma) and Roberto Ognibene (Pisa).

Jade Brisson

Title: Multiple tubular excisions and large Steklov eigenvalues

Abstract: For a given closed manifold, if we perform a small tubular excision of a submanifold with codimension greater than 2, then the Steklov eigenvalues diverge to infinity as the size of the tubular excision tends to zero. Moreover, the rate of divergence only depends only on the codimension. This construction is helpful to study different isoperimetric-type problems, such as maximizing the first perimeter-normalized Steklov eigenvalue among domains in a complete Riemannian manifold.

Lukas Bundrock

Title: The Robin Eigenvalue Problem in Exterior Domains

We consider the Robin eigenvalue problem in exterior domains, the complement of a compact domain. In contrast to the Robin eigenvalue problem in bounded domains, here there is a non-empty essential spectrum, so the first eigenvalue is not always discrete. 

On the one hand, we characterize, depending on the parameter of the boundary condition, when the first eigenvalue is discrete. On the other hand, we investigate which domain, among all bounded smooth domains with fixed volume, maximizes the first Robin eigenvalue in exterior domains. We show that the ball is a strict local maximizer, but not the global maximizer. Similar results can be obtained for the Steklov eigenvalue problem in exterior domains.

Davide Carazzato

Title: On the maximizers of a Wasserstein-type energy     

 Abstract: We study an energy defined through an optimal transport problem, which previously appeared in some models representing biological membranes. We prove that the maximizers coincide with a ball in many relevant cases, and this is done by means of a symmetrization technique. This talk is based on a joint work with Almut Burchard and Ihsan Topaloglu.

Francesco Chiacchio

Title: Weighted and unweighted symmetrization for nonlinear elliptic Robin problems. 

Abstract: We consider the p-Poisson equation with Robin boundary conditions, where the Robin parameter is a function. We provide various bounds for the solution to the problem under consideration using both classical Schwarz rearrangement and a suitable weighted symmetrization.  In the last case we also derive a Faber-Krahn type inequality. This talk is based on some joint papers with A. Alvino, V. Amato, A. Gentile, C. Nitsch, C. Trombetti. 

Marco Cicalese

Ttile: From crystals to Wulff shapes

Abstract:  We introduce the crystallization problem and its connection to the Wulff problem. We focus on the minimization of Heitman-Radin potential energies for configurations of N particles. We identify the asymptotic Wulff shapes through Gamma-convergence, we introduce the concept of fluctuation and discuss optimal fluctuations for quasiminimizers of the anisotropic perimeter giving as a corollary a short proof of the well-known N^{3/4} conjecture for minimizers on planar lattices. The technique combines the sharp quantitative Wulff inequality with a notion of quantitative closeness between discrete and continuum problems. We eventually focus on some recent result on the three dimensional case obtained in collaboration with Gian Paolo Leonardi (Trento) and Leonard Kreuz (TU Münich) and discuss some open problems.

Simone Cito

Title: Minimization of the Robin-Laplacian eigenvalues with perimeter constraint

Abstract: In this work we study the existence and some properties of the minimizers for the Robin-Laplacian eigenvalues with positive boundary parameter and perimeter constraint. We set our analysis in a relaxed framework: we consider sets of finite perimeter with possible inner fractures and we relax the definition of the eigenvalues accordingly, in such a way to consider the two-sided contribution of the inner cracks both in the perimeter constraint and in the Robin energy. By using this new approach we are able to prove the existence and the boundedness of minimizers for the relaxed problem and to approximate these minimizers via smooth sets. The talk is based on a forthcoming work in collaboration with A. Giacomini.

Bruno Colbois

Title: Geometric aspects of the ground state of magnetic Laplacians on domains of the plane.

Abstract: I will survey a few classical and recent results for the ground state (that is the first eigenvalue of the spectrum) of magnetic Laplacians on domains of the plane. I will consider the Aharonov-Bohm magnetic Laplacian (curvature zero) and the magnetic Laplacian with constant curvature. I will discuss isoperimetric inequalities and geometric bounds. I will not describe the different proofs, but I will take the time to mention different open questions.

The recent results are obtained in different papers in collaboration with Corentin Lena, Luigi Provenzano, Alessandro Savo.

Marc Dambrine

Title: Robust Shape Optimization

Francesco Della Pietra

Ttile: Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

Abstract: In this talk, I will discuss some optimal upper and lower bounds for the first Robin eigenvalue of the p-Laplacian operator, with Robin boundary conditions. These bounds are expressed in terms of geometrical quantities related to the domain, such as the volume, perimeter and inradius. I will also discuss the case where p approaches 1.

Sam Farrington

Title: Asymptotic behaviour of convex optimal spectral domains under diameter and perimeter constraint

Abstract: In this talk, we will consider minimising the k-th eigenvalue of the Laplacian over collections of convex domains with diameter/perimeter constraint under Dirichlet, Neumann and Zaremba boundary conditions. In particular, we will look at the asymptotic behaviour of sequences of such optimal domains, when they exist, as k goes to infinity in each case. Naïvely, one may assume that Weyl’s law and the isodiametric/isoperimetric inequalities inequality imply that such sequences converge to the solution of the isodiametric/isoperimetric problem over the given collection of domains as k goes to infinity. We will look at when this naïve assumption is true and present some new results in this direction for Neumann and Zaremba boundary conditions, starting in two dimensions and then going higher.

Ilaria Fragalà

Title. Riesz inequality for polygons: symmetry and symmetry breaking. 

Abstract. I will discuss counterparts of the classical Hardy-Littlewood and Riesz inequalities  

when the class of admissible domains is the family  of polygons with a fixed number of  sides. 

The latter corresponds to study the polygonal isoperimetric problem in nonlocal version.

Based on a joint work with Beniamin Bogosel and Dorin Bucur.

Ilias Ftouhi

Titre: About some shape optimzation problems motivated by urban planning

Abstract : In this talk, we will introduce the topic of shape optimization and some of its applications in real life problems. Then we will focus on some theoretical problems motivated by the following question: where should we place a park inside a given neighborhood and how should it be designed in order to make it the closest (in some relevant sense) to all the residents of the district? The talk is based on joint works in collaboration with Zakaria Fattah (ENSAM, Morocco) and Enrique Zuazua (FAU, Germany). 

Nunzia Gavitone

Title: On a Serrin-type Overdetermined Problem with Robin Boundary Conditions

Abstract: In this talk we will consider the torsion problem for the Laplace operator with Robin boundary condition and we will study an overdeterminated problem. In particular under suitable assumptions, we will prove the Serrin’s rigidity result. The results I will describe are contained in a joint work with Riccardo Molinarolo.

Maria Stella Gelli

Title: A mass optimization problem with convex cost. 

Abstract.  We study a mass optimization problem in the case of scalar state functions,

where instead of imposing a constraint on the total mass of the competitors, we penalize the classical compliance by a convex functional defined on the space of measures. We obtain a characterization of optimal solutions to the problem through a suitable PDE. This generalizes the case considered in the literature of a linear cost and applies to the optimization of a conductor where very low and very high conductivities have both a high cost, and then the study of nonlinear models becomes relevant. This is a  joint work with G. Buttazzo and D. Lučić.

Andrea Gentile

Title: Estimates for Robin p-Laplacian eigenvalues of convex sets with prescribed perimeter

Abstract: We will consider the shape optimization problem of minimizing/maximizing the first eigenvalue of the p-Laplace operator with Robin boundary conditions in the class of convex sets. In particular, when imposing a perimeter constraint, we will study the behavior of the eigenvalues as the boundary parameter beta varies in R. We prove an upper bound for the first Robin eigenvalue of the p- Laplacian with a positive boundary parameter and a quantitative version of the reverse 

Faber-Krahn type inequality for the first Robin eigenvalue of the p-Laplacian with negative boundary parameter, making use of a comparison argument obtained by means of inner parallel sets. 

Alexandre Girouard

Title: Isoperimetric-type inequalities and large Steklov spectral gap

I will discuss various isoperimetric-type inequalities for Steklov eigenvalues obtained in collaboration with Bruno Colbois, Ahmad El Soufi, and Katie Gittins. In order to understand how effective these inequalities are, it is useful to construct families of manifolds for which the Steklov spectral gap becomes arbitrarily large. In particular, I will describe work with Panagiotis Polymerakis (University of Thessaly) in which we use the Bakry-Emery Laplacian o obtain new examples of compact manifolds with arbitrarily large Steklov spectral gap.

Asma Hassannezhad 

Title: Steklov eigenvalues of pinched negatively curved manifolds

Abstract: The Steklov problem is an eigenvalue problem with spectral parameters on the boundary and is closely related to the Laplace eigenvalue problem. The Steklov eigenvalues in a bounded domain with a smooth boundary have the same asymptotic behaviour as the square root of Laplace eigenvalues of the boundary.  However, their geometric bounds can differ significantly, especially for the lower-order eigenvalues.  In this talk, we study the Steklov eigenvalue problem on a pinched negatively curved manifold of dimension at least three with a totally geodesic boundary. In this setting, we show that lower-order Steklov eigenvalues enjoy a similar geometric bound, akin to the lower eigenvalue bound for the first nonzero Laplacian eigenvalue on a closed manifold. The lower bounds we obtain are in terms of the volume of the manifold or area of the boundary, and the pinching constant. A key element in the proof is obtaining a tubular neighbourhood theorem for a totally geodesic hypersurface in negatively curved manifolds which can 

Antoine Henrot

Ttile: Two optimization problems for Neumann eigenvalues

Hedwig Keller

Title: Numerical Approximation of Optimal Convex Shapes in R^3 for an eigenvalue problem arising in optimal insulation:

Abstract: n the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in R^3.

A notion of discrete convexity allows for convex approximation with polyhedral domains. In the relaxation to discrete convex domains, the regularity of boundaries of convex domains is lost. We investigate the optimization of an eigenvalue problem arising in optimal insulation and prove the stability of the shape optimization algorithm under the constraint to discrete convex domains. An alternative is based on the recent observation that higher order finite elements can approximate convex functions conformally. As a second approach these results are used to approximate optimal convex domains with isoparametric convex domains.

Lorenzo Lamberti

Title: A regularity result for a class of free boundary problems

Abstract: The talk focuses on the study of a class of free boundary problems involving both bulk and interface energies. The bulk energy is of Dirichlet type, albeit of very general form, allowing the dependence from the unknown variable u and the position x. We employ the regularity theory of Λ-minimizers to study the regularity of the free interface. Mild assumptions concerning the dependence of the coefficients on x and u are made and are of Hölder type. The talk is based on a joint work with Luca Esposito.

Corentin Léna

Title: Bounds for Neumann eigenvalues of parallelograms and strips

Abstract: I will present sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of planar domains: parallelograms and strips. These results are deduced from  Rayleigh's principle using trial functions constructed from a suitable mapping of the domain onto the unit square.  They give in particular a new proof of an isoperimetric inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and I. Lucardesi and related to a conjecture of R. Laugesen and B. A. Siudeja.

Roméo Leylekian

Title: Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate

Abstract: The first eigenvalue of the Dirichlet bilaplacian shall be interpreted as the principal frequency of a vibrating plate with clamped boundary. In 1894, Rayleigh conjectured that, upon prescribing the area, the vibrating clamped plate with least principal frequency is circular. In 1995, Nadirashvili proved the Rayleigh Conjecture. Subsequently, Ashbaugh and Benguria proved the analogue of the conjecture in dimension 3. Since then, the conjecture has remained open in dimension d > 3. In this document, we contribute in answering the conjecture in high dimension under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti’s comparison principle, made possible after a fine study of the geometry of the eigenfunction’s nodal domains.

Alba Lia Masiello

Title: Sharp inequalities involving the Cheeger constant of planar convex sets

Abstract: We are interested in proving sharp inequalities  relating the Cheeger constant to different geometrical quantities, such as the area, the perimeter, the inradius, the circumradius, the minimal  width and the diameter. In particular, we provide new sharp bounds between these quantities for open, bounded and convex sets in the plane and we give new conjectures.

Eshete, Benyam Mebrate

Title: An Overdetermined Problem Related to the Finsler $p$-Laplacian

Abstract: In this talk, we consider the Finsler $p$-Laplacian torsion equation. The domain of the 

problem is bounded by a conical surface supporting a Neumann-type condition, and an unknown 

surface supporting both a Dirichlet and a Neumann condition. The case when the cone coincides 

with the punctured space is included. We show that the existence of a weak solution implies that 

the unknown surface lies on the boundary
of a Finsler-ball. Incidentally, some properties of the Finsler-Minkowski norms are proved here under

 mild smoothness assumptions.

This is a joint work with Prof. Antonio Greco.

Anna Mercaldo

Title: Isoperimetric sets for weighted twisted eigenvalues

Mickaël Nahon 

Title: Some recent results for Neumann eigenvalues of domains in the sphere.

Abstract: In this talk I will present some recent results about inequalities of Neumann eigenvalues for domains in the sphere in comparison with geodesic balls, with some new inequalities for the first and second non-trivial eigenvalues and a discussion of analytic counterexamples for the first eigenvalue using the structure of Bessel functions on the sphere. This is a collaboration with Dorin Bucur, Richard Laugesen and Eloi Martinet.

Carlo Nitsch

Ttile: Spectral Problems for the Infinite Laplacian

Abstract: In this talk, I'll discuss spectral problems related to the infinite Laplacian. I'll cover classical and recent results concerning Dirichlet, Neumann, and Robin boundary conditions. Additionally, I will outline some of the open questions in this field.

Alain Didier Noutchegueme

Title : Isoperimetric inequalities for Steklov transmission eigenvalues on surfaces.

Abstract : Consider a curve on a closed surface endowed with a Riemannian metric. The Steklov transmission problem is to find continuous functions which are harmonic away  from the curve, and such that the jump of the normal derivative across the curve is proportional to the value of the function. Such functions are called Steklov transmission eigenfunctions, and the corresponding proportionality coefficients are called Steklov transmission eigenvalues. We will discuss isoperimetric inequalities for these eigenvalues,  and highlight some similarities and differences compared to the usual Steklov case. The talk is based on a joint work with Mikhail Karpukhin (UCL).

Roberto Ognibene

Title: Spectral stability for the Neumann-Laplacian in domains with small holes

Abstract: The question of spectral stability for the Laplace operator in domains with small holes is a widely investigated topic. If Dirichlet boundary conditions are imposed, then the behavior of the perturbed spectrum as the holes shrink is quite well understood, and a main role is known to be played by the capacity of the holes. On the other hand, as far as Neumann boundary conditions are concerned, much less is known. In this talk I will address the latter case and present some recent results obtained in collaboration with V. Felli and L. Liverani, in which we sharply describe how fast the perturbed eigenvalues converge to the unperturbed ones, as the holes tend to disappear. Here, the main role is played by a geometric quantity resembling a notion of torsional rigidity of the holes.

Luigi Provenzano

Title: Geometry of the magnetic Steklov problem on Riemannian annuli.

Abstract: We consider the first two normalized Steklov eigenvalues of the magnetic Laplacian with zero mangetic field and flux $\nu$ on Riemannian annuli. We obtain sharp upper bounds for the eigenvalues and we discuss the geometry of the maximisers. Joint work with Alessandro Savo.

Edouard Oudet

Title: Minimal surfaces and Harmonic functions on finitely-connected tori 

Abstract: In this talk, we prove a Logarithmic Conjugation Theorem on finitely-connected tori. 

The theorem states that a harmonic function can be written as the real part of a function whose 

derivative is analytic and a finite sum of terms involving the logarithm of the modulus of a modified 

Weierstrass sigma function. We implement the method using arbitrary precision and use the result 

to find approximate solutions to the Laplace problem and Steklov eigenvalue problem. Using a 

posteriori estimation, we show that the solution of the Laplace problem on a torus with a few holes 

has error less than 10−100 using a few hundred degrees of freedom and the Steklov eigenvalues have 

similar error. In collaboration with C.-Y. Kao and B.Osting.

Iosif Polterovich

Title: Pólya's eigenvalue conjecture: some recent advances

Abstract: The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting

functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be

estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. The

conjecture is known to be true for domains which tile the Euclidean space, however it remains 

largely open in full generality. In the talk we will explain the motivation behind this conjecture

and discuss some recent advances, notably, the proof of Pólya’s conjecture for the 

disk. The talk is based on a joint work with Nikolay Filonov, Michael Levitin and David Sher.

Paolo Salani

Title: Anisotropic overdetermined problems

Abstract: I will give some news about an overdetermined problem for the Finsler Laplacian and discuss an overdetermined problem for a Monge-Ampèe type equation, modeled upon general anisotropic norms H in Rn. Similarly to what happens in the Euclidean case, when the solution is subject to both a homogeneous Dirichlet condition and a second boundary condition, designed on H, the domain (and the solution) must have a Wulff shape symmetry associated with H. The talk is based on a joint paper with Andrea Cianchi.

Giorgio Saracco

Title: Existence of minimizers of Cheeger's functional among convex sets

Abstract: Given a bounded, convex set Omega, it is well known that the ratio between the p-th power of its Cheeger constant and its first Dirichlet eigenvalue of the p-Laplacian is bounded from below. Parini first (for p=2) and Briani--Buttazzo--Prinari later (for general p) showed existence of minimizers of such a ratio in the 2d case and conjectured the same to hold true regardless of the dimension. We positively solve such conjecture combining a criterion by Ftouhi and cylindrical estimates on the Cheeger constant. If time allows, we shall discuss supremizing sequences in the 3d case.

Giacomo Vianello 

Title: A vertex-skipping property for almost-minimizing sets in convex containers

Abstract: given a convex set K in the Euclidean 3-space, we focus on the boundary behavior of an almost-minimizer E for the relative perimeter in K. We show that the closure of the internal boundary of E cannot contain vertex-type singularities of the boundary of K. To prove this result, we exploit a blow-up argument that reduces the problem to the instability of a suitable plane passing through the vertex of a conical container. One of the intermediate results, that for instance allows us to consider a larger class of almost-minimizers, is a boundary monotonicity formula valid under some mild, extra assumptions on K. These results are part of my PHD thesis and have been obtained in collaboration with my Supervisor Prof. Gian Paolo Leonardi.

Anna Chiara Zagati 

Title: Sharp geometric estimates for Sobolev-Poincaré constants involving the distance function

Abstract: On an open set, whose distance function satisfies some summability properties, we prove a lower bound for the sharp Sobolev-Poincaré constants in terms of the norm of the distance function in a suitable Lebesgue space. Moreover, on the class of convex sets, we find the sharp constant for such a lower bound, by generalizing a result shown by E. Makai in the planar case. Finally, we compare the sharp constants obtained in the class of convex sets with the optimal constants defined in other classes of open sets.

The results presented in this talk are obtained in collaboration with Lorenzo Brasco (University of Ferrara) and Francesca Prinari (University of Pisa).