Minimizing the so-called “Dirichlet energy” with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.
Keywords: Shape optimization, Dirichlet energy, existence/stability of optimal shapes
@article{COCV_2021__27_S1_A7_0,
author = {Henrot, Antoine and Mazari, Idriss and Privat, Yannick},
title = {Shape optimization of a {Dirichlet} type energy for semilinear elliptic partial differential equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020052},
mrnumber = {4222160},
zbl = {1467.49034},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020052/}
}
TY - JOUR AU - Henrot, Antoine AU - Mazari, Idriss AU - Privat, Yannick TI - Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020052/ DO - 10.1051/cocv/2020052 LA - en ID - COCV_2021__27_S1_A7_0 ER -
%0 Journal Article %A Henrot, Antoine %A Mazari, Idriss %A Privat, Yannick %T Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020052/ %R 10.1051/cocv/2020052 %G en %F COCV_2021__27_S1_A7_0
Henrot, Antoine; Mazari, Idriss; Privat, Yannick. Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. S6. doi: 10.1051/cocv/2020052
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