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SUMMARY:Shape optimization of light structures and the vanishing mass conj
ecture
DTSTART;VALUE=DATE-TIME:20220531T131500Z
DTEND;VALUE=DATE-TIME:20220531T141500Z
DTSTAMP;VALUE=DATE-TIME:20220810T082143Z
UID:indico-event-7645@indico.math.cnrs.fr
DESCRIPTION:We present rigorous results about the vanishing-mass limit of\
nthe classical problem to find a shape with minimal elastic compliance. Co
ntrary to all previous results in the mathematical literature\, which util
ize a soft mass constraint by introducing a Lagrange multiplier\, we here
consider\nthe hard mass constraint. Our results are the first to establish
the convergence of approximately optimal shapes of (exact) size $\\vareps
ilon\\to0$ 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 conject
ured by BouchittÃ© 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 oscil
lation of (optimal) lower-dimensional structures. Its appearance is surpri
sing since the integrand in the original compliance is just the Euclidean
norm and the non-convexity of the problem is not immediately obvious. In f
act\, it is the interaction of the\nmass constraint with the requirement o
f attaining the loading (in the form of a divergence-constraint) that give
s rise to this new integrand. Our proofs rest on compensated compactness a
rguments applied to an explicit family of (symmetric) divquasiconvex quadr
atic forms\, computations involving the Hashin-Shtrikman bounds for the Ko
hn-Strang integrand\, and the characterization of limit minimizers due to
BouchittÃ© & Buttazzo.\n\nThis is a joint work with J.-F. Babadjian and F.
Rindler.\n\nhttps://indico.math.cnrs.fr/event/7645/
LOCATION:ICJ Fokko
URL:https://indico.math.cnrs.fr/event/7645/
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