[Dottorcomp] Seminari di Matematica Applicata. Martedì 1 febbraio, Flaviana Iurlano

[Stefano Lisini]

Seminari di Matematica Applicata, Dipartimento di Matematica "F.
Casorati" e Istituto del CNR IMATI "E. Magenes" di Pavia.

Martedì 1 febbraio 2022, alle ore 14 precise, presso l'aula Beltrami
del Dipartimento di Matematica,

Flaviana Iurlano (Laboratoire Jacques-Louis Lions, Sorbonne
Université, Paris) terrà un seminario dal titolo:

Shape optimization of light structures and the vanishing mass conjecture

Il seminario verrà anche trasmesso in diretta su zoom al link

https://us02web.zoom.us/j/87174718701?pwd=SmdkTlkzcHJoVEZYd2IrMGs4anZUUT09

Meeting ID: 871 7471 8701
Passcode: 630697

Abstract.
We present rigorous results about the vanishing-mass limit of the
classical problem to find a shape with minimal elastic compliance.
Contrary to all previous results in the mathematical literature, which
utilize a soft mass constraint by introducing a Lagrange multiplier,
we here consider the hard mass constraint. Our results are the first
to establish the convergence of approximately optimal shapes of
(exact) size ε→0 to a limit generalized shape represented by a
(possibly diffuse) probability measure. This limit generalized shape
is a minimizer of the limit compliance, which involves a new
integrand, namely the one conjectured by Bouchitté in 2001 and
predicted heuristically before in works of Allaire & Kohn and Kohn &
Strang from the 1980s and 1990s. This integrand gives the energy of
the limit generalized shape understood as a fine oscillation of
(optimal) lower-dimensional structures. Its appearance is surprising
since the integrand in the original compliance is just the Euclidean
norm and the non-convexity of the problem is not immediately obvious.
In fact, it is the interaction of the mass constraint with the
requirement of attaining the loading (in the form of a
divergence-constraint) that gives rise to this new integrand. Our
proofs rest on compensated compactness arguments applied to an
explicit family of (symmetric) divquasiconvex quadratic forms,
computations involving the Hashin-Shtrikman bounds for the Kohn-Strang
integrand, and the characterization of limit minimizers due to
Bouchitté & Buttazzo. This is a joint work with J.-F. Babadjian and F.
Rindler.
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Pagina web dei Seminari di Matematica Applicata
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