# Seminario di Calcolo delle Variazioni & Equazioni alle Derivate Parziali

I seminari si tengono di norma di venerdì alle ore 14:30 nella Sala Conferenze "Franco Tricerri" del Dipartimento di Matematica e Informatica "Ulisse Dini" (Viale Morgagni 67/A).
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A.A. 2018 / 2019

--::: PROSSIMO SEMINARIO: :::--

martedì 11 giugno

14:30

 Roberto Alicandro (U. Cassino) Derivation of linear elasticity from atomistic energies with multiple wells

Abstract: Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth employing the geometric rigidity estimate of Friesecke, James, and Mueller. Such a result was later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this talk we present the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. The singular perturbation can be interpreted in terms of interactions beyond nearest neighbours in an atomistic model.

----:: Seminari in programma: ::----

17 giugno

14:30

 Matthias Eller (Georgetown University, DC, USA) On initial boundary value problems for hyperbolic systems

Abstract: Boundary value problems for hyperbolic systems of first order will be discussed. There are two competing theories. One for symmetric hyperbolic systems which can be traced back to Friedrichs (1954) and another one for strictly hyperbolic system which goes back to Kreiss (1970). While these theories are also of importance for nonlinear problem, this talk will focus on linear problems in a space-time cylinder. We will formulate optimal statements with respect to the regularity of the coefficients and the regularity of the boundary. Strictly dissipative boundary conditions, conservative boundary conditions, and the condition by Kreiss and Sakamoto will be discussed.

3 luglio

14:30 e 15:30

 Flaviana Iurlano (Laboratoire Jacques-Louis Lions) Concentration versus oscillation effects in brittle damage Sergio Conti (Universitaet Bonn) Dislocation microstructures and strain-gradient plasiticity with a single active slip plane

Abstract Iurlano: This work is concerned with an asymptotic analysis, in the sense of $\Gamma$-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order $\varepsilon$, the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at $\varepsilon$ fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as $\varepsilon\to 0$, concentration of damage and worsening of the elastic properties are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at $\varepsilon$ fixed) to being linear (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. In particular, the interaction of homogenization with singularity formation in the framework of linearized elasticity appears to not have been considered in the literature so far. We explicitly identify the $\Gamma$-limit in two and three dimensions for isotropic Hooke tensors. The expression of the limit effective energy turns out to be of Hencky plasticity type. We further consider the regime where the divergence remains square-integrable in the limit, which leads to a Tresca-type model.

Abstract Conti

----:: Seminari passati: ::----

31 maggio

14:30 e 15:30

 Gero Friesecke (Technische Universitat Munchen) Non-existence of minimizers in Monge optimal transport Michael Roysdon (Kent State University) Rogers-Shephard type inequalities and generalizations

Abstract Friesecke: Optimal transport (OT) problems arise not just by contemplating how to transport a pile of sand efficiently into a hole (Monge 1781), but also in fluid dynamics, PDE, economics, statistics, machine learning, and - recently (see [1] and parallel work of De Pascale, Buttazzo and Gori-Giorgi) - electronic structure. It is well known that the Monge ansatz may fail in continuous two-marginal OT. I explain [2] that this effect already occurs for finite assignment problems with N=3 marginals, l=3 'sites', and symmetric pairwise costs, with the values for N and l both being optimal. Our counterexample is a transparent consequence of the convex geometry of the set of symmetric Kantorovich plans. By superposition, our example gives rise to a countinuous one, where failure of the Monge ansatz manifests itself as nonattainment and formation of 'microstructure'. In the simplified setting of discretized optimal transport, I will present a new ''Quasi-Monge'' ansatz [3] for symmetric N-marginal problems which - unlike the Monge ansatz - always yields the minimum Kantorovich cost but whose computational complexity scales linearly instead of exponentially with N. This result is motivated by applications to electronic structure. In this context N equals the number of particles, explaining the interest in large N. [1] C.Cotar, G.F., C.Klueppelberg, Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math. 66, 548-599, 2013 [2] G.Friesecke, A simple counterexample to the Monge ansatz in multi-marginal optimal transport, convex geometry of the set of Kantorovich plans, and the Frenkel-Kontorova model, 2018 https://arxiv.org/abs/1808.04318 [3] G.Friesecke, D.Vögler, Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, SIAM J. Math. Analysis Vol. 50 No. 4, 3996-4019, 2018

Abstract Roysdon

24 maggio

14:30

 Paolo Vannucci (U. Versailles) The polar formalism in analysis and optimal design of anisotropic structures

Abstract: The polar formalism was introduced by G. Verchery as early as 1979 as a method for finding the invariants of a planar tensor of any rank. With this method, a tensor is represented by invariants and angles; as such, this approach is particularly suited for representing anisotropic problems because it allow to introduce explicitly, on one hand, the intrinsic properties of the material with regard to a given physical phenomenon, through the invariants, and, on the other hand, geometrical parameters determining the direction: angles. The polar formalism is particularly useful and effective in two circumstances: the study of some theoretical problems, namely linked to the symmetries, that are introduced in a natural and direct, algebraic way, and the design problems concerning anisotropic structures. In the talk, we show first the basic elements of the polar formalism, then we focus on some theoretical considerations and problems solved thanks to the polar formalism and finally we show some modern design problems of anisotropic structures formulated as optimization problems in the design space of the polar parameters.

17 maggio

11:00-17:30

 MathAnalysis@UniFIPISI Incontri di Analisi Matematica tra Firenze, Pisa e Siena

La registrazione, il programma dettagliato ed ulteriori informazioni si trovano al seguente indirizzo: http://events.dimai.unifi.it/FIPISI

10 maggio

14:30

AULA 2

 Seminario di Geometria, Analisi Complessa, Calcolo delle Variazioni ed Equazioni alle Derivate Parziali Daniele Valtorta (U. Zuerich) Quantitative stratification and applications

Abstract: In this talk we make a brief overview of the Quantitative stratification technique and its applications to study the singular sets of various geometric objects (harmonic maps, minimal surfaces, Manifolds with Ricci bounds and obstacle problems), and comparing the different problems that arise with different examples.

3 maggio

14:30 e 15:30

 Keith Miller (University of California, Berkeley) A backprojection kernel for very-wide-angle PET (Positron Emission Tomography) Zdenek Mihula (Charles University, Prague) Reduction principles and their applications in Sobolev spaces

Abstract Miller: Abstract

Abstract Mihula: We introduce the idea of so-called reduction principles. We present it within the framework of Sobolev spaces and show not only how these principles can be used for finding optimal spaces in Sobolev-type inequalities, but also how these principles help us to establish compactness results.

12 aprile

14:30

 Alberto Farina (Université Picardie Jules Verne) Monotonicity and symmetry of solutions to a non-cooperative system of Gross-Pitaevskii-type

Abstract: Abstract

5 aprile

14:30

 Giuliano Lazzaroni (U. Firenze) On the 1d wave equation in time-dependent domains and the problem of dynamic fracture

Abstract: Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is possibly degenerate at the initial time. First we prove existence for the wave equation when the evolution of the domain is given; in the more general case, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of dynamic fracture and crack initiation.

15 marzo

14:30

 Philipp Kniefacz (Vienna University of Technology) Sharp Sobolev Inequalities via Projection Averages

Abstract: In this talk we present a family of sharp Sobolev-type inequalities obtained from averages of the length of $i$-dimensional projections of the gradient of a function. This family has both the classical Sobolev inequality (for $i = n$) and the affine Sobolev-Zhang inequality (for $i = 1$) as special cases as well as a recently obtained Sobolev inequality of Haberl and Schuster (for $i = n - 1$). Moreover, we identify the strongest member in our family of analytic inequalities which turns out to be the only affine invariant one among them. (joint work with F.E. Schuster)
1 marzo

14:30

 Elisabetta Chiodaroli (U. Pisa) On the energy equality for the 3D Navier-Stokes equations

Abstract: In this talk we consider weak solutions to the 3D Navier-Stokes equations in a smooth domain with Dirichlet conditions and we discuss the validity of the energy equality in this class. We prove some new conditions for energy conservation and we compare them with classical and more recent results of the existing literature, in particular in view of the famous Onsager conjecture. Finally, we analyze the problem of energy conservation for very weak solutions. This is a joint work with Luigi C. Berselli.

22 febbraio

14:30

 Guido De Philippis (SISSA) Regularity for a Model of Charged Droplets

Abstract: First I will review some features of the mathematical modelization of charged droplets. I will then focus on a model, proposed by Muratov and Novaga, which takes into account the regularization effect due to the screening of free counterions in the droplet. In particular I will present a partial regularity result for minimizers and I will present some open problems. This is joint work with J. Hirsch e G. Vescovo.

25 gennaio

14:30

 Marco Spadini (U. Firenze) Un modello per l'HIV - Struttura dell'insieme delle soluzioni periodiche

Abstract: In questo lavoro, in collaborazione con Luca Bisconti, prendiamo in esame un modello matematico ben noto del meccanismo di trasmissione del virus dell'HIV tra cellule del tessuto linfatico. Senza entrare nel merito del meccanismo biologico, il modello si riduce ad un sistema di equazioni con ritardo (infinito). Introducendo una perturbazione periodica (corrispondente, per esempio, a fasi di cura o reinfezione esterna) dipendente da un parametro che ne determina l'intensità, ci concentrimo sull'insieme delle soluzioni periodiche del modello così modificato, studiando la biforcazione e le proprietà di rami di soluzioni periodiche non banali.

18 gennaio 2019

14:30

 Nicolas Van Goethem (Universidade de Lisboa) Variational evolution of dislocations in single crystals

Abstract: In this talk I will present some recent results about the mathematical modelling of line defects in single crystals and the analysis of dislocation singularities. Existence of minimizers to a problem of finite elasticity with dislocations will be discussed as well as the variational evolution of dislocation networks in single crystals. This is a joint work with Riccardo Scala since 2012.

14 dicembre 2018

14:30

 Simone Di Marino (INdAM, SNS) Extensions theorems for Lipschitz functions

Abstract: Given two metric spaces $X \subseteq Y$ and a Lipschitz function $f:X \to Z$, one asks if it is possible to find a function $g:Y \to Z$ such that $g$ arees with $f$ on $X$ and still $g$ is Lipschitz. The problem is quite old and several question has been already answered. I will talk about two contribution: in the first one (in collaboration with F. Stra and E. Brué) we find with a general linear extension operator, which will solve the case when $Z$ is a general Banach space. The only hypotesis will be that $X$ is a doubling space. The second contribution (in collaboration A. Pratelli and N. Gigli) instead is in the case $Z=\mathbb{R}$, but general $X$ and $Y$ (it has applications in the theory of Sobolev spaces in metric measure spaces). In this case of course the MacShane construction gives an extension which moreover preserves the Lipschitz constant; however the asymptotic Lipschitz constant could (and in fact) degenerate. We prove that we can construct an extension which preserves the asymptotic Lipschitz constant on the whole $X$, at the price of losing something in the global Lipschitz constant.
7 dicembre

14:30 e 15:15

14:30 Giovanni Bellettini (U. Siena)
Nuovi risultati sul rilassamento del funzionale dell'area di grafici di mappe discontinue dal piano in sé
 15:15 Emanuele Paolini (U. Pisa) Cluster minimi nel piano

Abstract Bellettini: Illustrerò alcuni risultati recenti sul rilassamento del funzionale dell'area per grafici due dimensionali e due codimensionali, per mappe discontinue. Nel caso di mappe a tre valori, si metter&arave; in evidenza la relazione del problema di rilassamento con alcuni problemi di tipo Plateau cartesiano che coinvolgono tre superfici tra loro accoppiate attraverso una condizione di Dirichlet con un punto triplo. I risultati sopra menzionati sono il frutto di una collaborazione in corso con A. Elshorbagy (SISSA, Trieste), R. Scala (Univ. Roma La Sapienza) e M. Paolini (Univ. Cattolica di Brescia).

Abstract Paolini: Parleremo del problema di racchiudere e separare N regioni di area fissata nel piano utilizzando il minimo perimetro. In particolare parleremo di un risultato recentemente ottenuto in collaborazione con V.M. Tortorelli sulla simmetria dei cluster formati da 4 regioni di uguale area.
30 novembre

14:30

 Riccardo Adami (Politecnico di Torino) The problem of the Ground States for the Nonlinear Schroedinger Equation on Metric Graphs: the two-dimensional grid

Abstract: Motivated by the study of the Ground State of Bose-Einstein Condensate on complicated structures, the problem of minimizing the NLS energy on metric graphs has been recently addressed in a sistematic way, exploring the cases of finite graphs with halflines, periodic graphs and trees. We give results for the two-dimensional grid, both in the Lˆ2 subcritical and critical cases. We show that the interdimensional structure of the grid, gives rise to a phenomenon called dimensional crossover, involving a continuum of Lˆ2-critical nonlinearity. This is a joint project with Simone Dovetta, Enrico Serra, Lorenzo Tentarelli, and Paolo Tilli.
23 novembre

14:30

 Frank Duzaar (U. Erlangen) Higher integrability for porous medium type systems

Abstract: In this talk we report on recent developments concerning the higher integrability of the spatial gradient to porous medium type systems of the form ∂_t u - Δ (|u|^{m-1}u) = Div F.
16 novembre

14:30

 Lubos Pick (Charles University Prague) Higher-order Sobolev embeedings, isoperimetric problem and Frostman measures

Abstract: We will survey both classical and modern techniques and results on higher-order Sobolev embeddings and trace embeddings. We shall focus on sharpness of function spaces in such embeddings obtained via reduction principles.
26 ottobre

14h30

 Marco Barchiesi (U. Napoli Federico II) Stability of the Gaussian Isoperimetric Inequality

Abstract: I will present an analysis of the sets that minimize the gaussian perimeter plus the norm of the barycenter. These two terms are in competition, and in general the solutions are not the half-spaces. In fact we prove that when the volume is close to one, the solutions are the strips centered in the origin. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one. Co-Author: Vesa Julin
19 ottobre

14h30

 Andrea Marchese (U. Pavia) Local minimality of strictly stable extremal submanifolds

Abstract: I will discuss a recent extension of a result by Brian White, who proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in its homology class, if the minimization problem is restricted to a certain geodesic tubular neighborhood of the submanifold. We replace the tubular neighborhood with one induced by the flat distance of integral currents and we provide quantitative estimates. The proof is based on the so called "selection principle", which, via a penalization technique, allows us to recast the problem in the class of graphs, exploiting the regularity theory for almost minimizers. Joint work with D. Inauen (Zurich).
Lunedì 24 settembre

15:30

 Jan Kristensen (University of Oxford) Regularity for minimizers of variational integrals of (1,p) growth

Abstract: In this talk I present some recent results on the partial regularity of BV minimizers for strongly quasiconvex variational integrals. The focus will be on the case of integrands satisfying linear growth conditions, but for the sake of illustrating the flexibility of a key argument, I'll show how it also applies in the (1,p) growth case when p is less than n/(n-1). The results are part of joint work with Franz Gmeineder (Bonn)

Organizzatori: Chiara Bianchini, Matteo Focardi.
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