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 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: 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 strain-gradient terms often used in mechanical applications. The second one is a nonlocal effective energy representing the far-field 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 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.

marted&igrave 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.

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.VÃ¶gler, 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