In this article, we are interested in shape optimization problems where the functional is defined on the boundary of the domain, involving the geometry of the associated hypersurface (normal vector n, scalar mean curvature H) and the boundary values of the solution uΩ related to the Laplacian posed on the inner domain Ω enclosed by the shape. For this purpose, given ε > 0 and a large hold-all B ⊂ ℝ$$, n ≥ 2, we consider the class $$ of admissible shapes Ω ⊂ B satisfying an ε-ball condition. The main contribution of this paper is to prove the existence of a minimizer in this class for problems of the form $$. We assume the continuity of j in the set of variables, convexity in the last variable, and quadratic growth for the first two variables. Then, we give various applications such as existence results for the configuration of fluid membranes or vesicles, the optimization of wing profiles, and the inverse obstacle problem.
Mots-clés : Shape optimization, uniform ball condition, elliptic partial differential equations, geometric functionals, existence theory, boundary shape identification problems
@article{COCV_2020__26_1_A108_0,
author = {Dalphin, J\'er\'emy},
title = {Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2020026},
mrnumber = {4185053},
zbl = {1459.49026},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2020026/}
}
TY - JOUR AU - Dalphin, Jérémy TI - Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2020026/ DO - 10.1051/cocv/2020026 LA - en ID - COCV_2020__26_1_A108_0 ER -
%0 Journal Article %A Dalphin, Jérémy %T Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2020026/ %R 10.1051/cocv/2020026 %G en %F COCV_2020__26_1_A108_0
Dalphin, Jérémy. Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 108. doi : 10.1051/cocv/2020026. http://www.numdam.org/articles/10.1051/cocv/2020026/
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