Optimal shapes for general integral functionals
Buttazzo, Giuseppe1; Shrivastava, Harish1
1 Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56126 Pisa, ITALY
Annales Henri Lebesgue, Volume 3 (2020), pp. 261-272.

We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain Ω that varies over all subdomains of a given bounded domain D of ℝ d . We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.

On considère des problèmes d’optimisation de forme pour des fonctionnelles intégrales générales, parmi les domaines Ω parcourant tous les sous-domaines d’un domaine borné D donné de ℝ d . Nous montrons de manière assez élémentaire l’existence d’une solution qui en général est un ensemble quasi ouvert. Sous des conditions très faibles, nous prouvons que le domaine optimal est en fait ouvert et de périmètre fini. Des contre-exemples montrent que ce n’est pas toujours le cas.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/ahl.31
Classification: 49Q10, 49A15, 49A50, 35J20, 35D10
Keywords: shape optimization, quasi open sets, finite perimeter, integral functionals.
Buttazzo, Giuseppe 1; Shrivastava, Harish 1

1 Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56126 Pisa, ITALY 
@article{AHL_2020__3__261_0,
     author = {Buttazzo, Giuseppe and Shrivastava, Harish},
     title = {Optimal shapes for general integral functionals},
     journal = {Annales Henri Lebesgue},
     pages = {261--272},
     publisher = {\'ENS Rennes},
     volume = {3},
     year = {2020},
     doi = {10.5802/ahl.31},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/ahl.31/}
}
TY  - JOUR
AU  - Buttazzo, Giuseppe
AU  - Shrivastava, Harish
TI  - Optimal shapes for general integral functionals
JO  - Annales Henri Lebesgue
PY  - 2020
SP  - 261
EP  - 272
VL  - 3
PB  - ÉNS Rennes
UR  - http://archive.numdam.org/articles/10.5802/ahl.31/
DO  - 10.5802/ahl.31
LA  - en
ID  - AHL_2020__3__261_0
ER  - 
%0 Journal Article
%A Buttazzo, Giuseppe
%A Shrivastava, Harish
%T Optimal shapes for general integral functionals
%J Annales Henri Lebesgue
%D 2020
%P 261-272
%V 3
%I ÉNS Rennes
%U http://archive.numdam.org/articles/10.5802/ahl.31/
%R 10.5802/ahl.31
%G en
%F AHL_2020__3__261_0
Buttazzo, Giuseppe; Shrivastava, Harish. Optimal shapes for general integral functionals. Annales Henri Lebesgue, Volume 3 (2020), pp. 261-272. doi : 10.5802/ahl.31. http://archive.numdam.org/articles/10.5802/ahl.31/

[BB05] Bucur, Dorin; Buttazzo, Giuseppe Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations, 65, Birkhäuser, 2005 | MR | Zbl

[BDM93] Buttazzo, Giuseppe; Dal Maso, Gianni An existence result for a class of shape optimization problems, Arch. Ration. Mech. Anal., Volume 122 (1993) no. 2, pp. 183-195 | DOI | MR | Zbl

[BHP05] Briançon, Tanguy; Hayouni, Mohamed; Pierre, Michel Lipschitz continuity of state functions in some optimal shaping, Calc. Var. Partial Differ. Equ., Volume 23 (2005) no. 1, pp. 13-32 | DOI | MR | Zbl

[Bri04] Briançon, Tanguy Regularity of optimal shapes for the DirichletÕs energy with volume constraint, ESAIM, Control Optim. Calc. Var., Volume 10 (2004) no. 1, pp. 99-122 | DOI | MR | Zbl

[Buc12] Bucur, Dorin Minimization of the k-th eigenvalue of the Dirichlet Laplacian, Arch. Ration. Mech. Anal., Volume 206 (2012) no. 3, pp. 1073-1083 | DOI | MR | Zbl

[But89] Buttazzo, Giuseppe Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Reseqrch Notes in Mathematics Series, 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989 | MR | Zbl

[GG84] Giaquinta, Mariano; Giusti, Enrico Quasi-minima, Ann. Inst. Henri Poincaré Anal. Non Linéaire, Volume 1 (1984) no. 2, pp. 7-107 | Numdam | Zbl

[Giu03] Giusti, Enrico Direct Methods in the Calculus of Variations, World Scientific Publishing, Singapore, 2003 | Zbl

[Hay99] Hayouni, Mohammed Lipschitz continuity of the state function in a shape optimization problem, J. Conv. Anal., Volume 6 (1999) no. 1, pp. 71-90 | MR | Zbl

[HM05] Henrot, Antoine; Michel, Pierre Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications, 48, Springer, 2005 | Zbl

[Maz11] Maz’ya, Vladimir G. Sobolev Spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften, 342, Springer, 2011 | MR | Zbl

[Zie89] Ziemer, William P. Weakly Differentiable Functions, Springer, 1989 | Zbl

Cited by Sources: