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Stability in shape optimization with second variation

Lamboley, Jimmy; Dambrine, Marc (2019), Stability in shape optimization with second variation, Journal of Differential Equations, 267, 5, p. 3009-3045. 10.1016/j.jde.2019.03.033

Type
Article accepté pour publication ou publié
External document link
https://hal.science/hal-01073089
Date
2019
Journal name
Journal of Differential Equations
Volume
267
Number
5
Publisher
Elsevier
Published in
Paris
Pages
3009-3045
Publication identifier
Metadata
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Author(s)
Lamboley, Jimmy
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Institut de Mathématiques de Jussieu - Paris Rive Gauche [IMJ-PRG (UMR_7586)]
Dambrine, Marc
Laboratoire de Mathématiques et de leurs Applications [Pau] [LMAP]
Abstract (EN)
We are interested in the question of stability in the field of shape optimization. Precisely, we prove that under structural assumptions on the hessian of the considered shape functions, the necessary second order minimality conditions imply that the shape hessian is coercive for a given norm. Moreover, under an additional continuity condition for the second order derivatives, we derive precise optimality results in the class of regular perturbations of a domain. These conditions are quite general and are satisfied for the volume, the perimeter, the torsional rigidity and the first Dirichlet eigenvalue of the Laplace operator. As an application, we provide non trivial examples of inequalities obtained in this way.
Subjects / Keywords
shape optimization; isoperimetric inequalities; stability in shape optimization; second order sensitivity