
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
3 luglio 14:30 e 15:30 
Flaviana Iurlano (Laboratoire JacquesLouis Lions)

Concentration versus oscillation effects in brittle damage
 Sergio Conti (Universitaet Bonn)

Dislocation microstructures and straingradient 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 squareintegrable in the limit, which leads to a Trescatype model.
Abstract Conti: We consider a model for dislocations in a three dimensional crystal, which are restricted
to move in a single plane. We derive, within the framework of $\Gamma$convergence,
a reduced model in the limit of small lattice spacing. The limiting model contains two terms.
The first one is a continuous energy with linear growth, which depends
on a measure which characterizes the macroscopic dislocation density; this term corresponds to the
straingradient terms often used in mechanical applications. The second one is a
nonlocal effective energy representing the farfield interaction between
dislocations, it is quadratic and depends on the elastic strain. Both arise naturally as scaling
limits of the nonlocal elastic interaction. Relaxation and formation of microstructures at intermediate scales
are automatically incorporated in the limiting procedure.
This talk is based on joint work with Adriana Garroni and Stefan Mueller.
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 spacetime 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.
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 Gammaconvergence. This was first done in the case of onewell energies with superquadratic 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 multiwell 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 multiwell 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.
31 maggio 14:30 e 15:30 
Gero Friesecke (Technische Universitat Munchen)

Nonexistence of minimizers in Monge optimal transport
 Michael Roysdon (Kent State University)

RogersShephard 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 GoriGiorgi)  electronic structure.
It is well known that the Monge ansatz may fail in continuous twomarginal 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 ''QuasiMonge'' ansatz [3] for symmetric Nmarginal 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, 548599, 2013
[2] G.Friesecke, A simple counterexample to the Monge ansatz in multimarginal optimal transport, convex geometry of the set of Kantorovich plans, and the FrenkelKontorova model, 2018 https://arxiv.org/abs/1808.04318
[3] G.Friesecke, D.VÃ¶gler, Breaking the curse of dimension in multimarginal Kantorovich optimal transport on finite state spaces, SIAM J. Math. Analysis Vol. 50 No. 4, 39964019, 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 11:0017:30 
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 verywideangle 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 socalled 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 Sobolevtype 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 noncooperative system of GrossPitaevskiitype


Abstract: Abstract
5 aprile 14:30 
Giuliano Lazzaroni (U. Firenze) 
On the 1d wave equation in timedependent domains and the problem of dynamic fracture


Abstract: Motivated by a debonding model for a thin film peeled from a substrate, we analyse the onedimensional wave equation, in a timedependent 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 Sobolevtype 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 SobolevZhang 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 NavierStokes equations


Abstract:
In this talk we consider weak solutions to the 3D NavierStokes 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 cosÃ¬ 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 twodimensional grid 

Abstract:
Motivated by the study of the Ground State of BoseEinstein 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 twodimensional grid, both in the LË2 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 LË2critical 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^{m1}u) = Div F.
16 novembre 14:30 
Lubos Pick (Charles University Prague) 
Higherorder Sobolev embeedings, isoperimetric problem and Frostman measures 

Abstract: We will survey both classical and modern techniques and results on higherorder 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 halfspaces. 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.
CoAuthor: 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/(n1).
The results are part of joint work with Franz Gmeineder (Bonn)
Organizzatori: Chiara Bianchini, Matteo Focardi.
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