Abstracts

Elisa Davoli (Universität Wien)

Two-well rigidity  and multidimensional sharp-interface limits for solid-solid phase transition

In this talk we establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid-solid phase transition  in any arbitrary space dimensions, under a suitable anisotropic penaliyation of second variations. By means of Gamma-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the size of he interfaces between the two phases.

Lucia De Luca (Universita` di Pisa)

A minimization approach to the wave equation on time-dependent domains.
We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.
Joint work with Gianni Dal Maso (SISSA).

Maria Stella Gelli (Universita` di Pisa)

Intrinsic distances and relaxation of $L^\infty$-functionals.
Given a  supremal functional of the form $F(u)= \supess_{x \in \Omega} f(x,Du(x))$
where $\Omega\subseteq \R^N$ is a regular bounded open set,  $u\in \wi$, under a mild assumption on the sublevel sets  of F, we give a description of the sublevel sets of its lower semicontinuous envelope with respect to the weak* topology in terms of the level sets of suitable difference quotients. This result is instrumental to show that
the  lower semicontinuous envelopes of F  with respect to the weak* topology, the weak* convergence and the uniform convergence  are level convex (i.e. they  have convex sub-level sets). The proof of these results relies both on a deep  analysis of  the intrinsic distances  associated to F and on a careful use of variational tools such as Gamma-convergence.

Flaviana Iurlano (LJLL - Paris)

Concentration analysis of brittle damage

This talk is concerned with an asymptotic analysis of a variational model of brittle damage, when the damaged zone concentrates into a set of zero Lebesgue measure, and, at the same time, the stiffness of the damaged material becomes arbitrarily small. In a particular non-trivial regime, concentration leads to a limit energy
with linear growth as typically encountered in plasticity. I will show that, while the singular part of the limit energy can be easily described, the identification of the bulk part of the limit energy requires a subtler analysis of the concentration properties of the displacements.                                                                                              This is an ongoing work with J.F. Babadjian and F. Rindler.

Dorothee Knees (Universität Kassel)

Discussion of different notions of solutions and time-discretization schemes for rate-independent damage models

It is well known that rate-independent systems involving nonconvex stored energy functionals in general do not allow for time-continuous solutions even if the given data are smooth in time. Several solution concepts are proposed to deal with these discontinuities, among them the meanwhile classical global energetic approach and the more recent vanishing viscosity approach. Both approaches generate solutions with a well characterized jump behavior. However, the solution concepts are not equivalent. In this context, numerical discretization schemes are needed that efficiently and reliably approximate directly that type of solution that one is interested in. For instance, in the vanishing viscosity context it is reasonable to couple the viscosity parameter with the

time-step size. The aim of this lecture is to discuss different types of solutions for rate-independent systems, to propose suitable time-discretization schemes, to study their convergence and to characterize as detailed as possible the limit curves as the

discretization parameters tend to zero. The talk relies on joint work with

Riccarda Rossi (Brescia), Chiara Zanini (Torino) and Matteo Negri (Pavia).

[1] D.Knees and M.Negri, Convergence of alternate minimization schemes for

phase field fracture and damage, Mathematical Models and Methods in Applied

Sciences, vol.27(9), pp. 1743-1794, 2017.

[2] D.Knees, R.Rossi, C.Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models and Methods in Applied Sciences, vol.23(04), pp. 565-616, 2013.

[3] D.Knees, R.Rossi, C.Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlinear Analysis Series B: Real World Applications, vol. 24, pp. 126-162, 2015.

Carolin Kreisbeck (Universiteit Utrecht)