Geodesic
Shape derivative of the Cheeger constant
Parini, Enea1 ; Saintier, Nicolas2, 3
1 LATP, Aix-Marseille Université, 39 rue Joliot Curie, 13453 Marseille cedex 13, France.
2 Instituto de Ciencias, University Nac. Gral Sarmiento, J. M. Gutierrez 1150, C.P. 1613 Los Polvorines Pcia de Bs. As, Argentina
3 Dpto Matemática, FCEyN, University de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 348-358

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This paper deals with the existence of the shape derivative of the Cheeger constant h 1 (Ω) of a bounded domain Ω. We prove that if Ω admits a unique Cheeger set, then the shape derivative of h 1 (Ω) exists, and we provide an explicit formula. A counter-example shows that the shape derivative may not exist without the uniqueness assumption.

Reçu le :
DOI : 10.1051/cocv/2014018
Classification : 49Q10, 49Q20
Keywords: Shape derivative, CHEEGER constant, 1-Laplacian

Parini, Enea 1 ; Saintier, Nicolas 2, 3

1 LATP, Aix-Marseille Université, 39 rue Joliot Curie, 13453 Marseille cedex 13, France.
2 Instituto de Ciencias, University Nac. Gral Sarmiento, J. M. Gutierrez 1150, C.P. 1613 Los Polvorines Pcia de Bs. As, Argentina
3 Dpto Matemática, FCEyN, University de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina. 
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     author = {Parini, Enea and Saintier, Nicolas},
     title = {Shape derivative of the {Cheeger} constant},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {348--358},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {2},
     year = {2015},
     doi = {10.1051/cocv/2014018},
     mrnumber = {3348401},
     zbl = {1315.49018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014018/}
}
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Parini, Enea; Saintier, Nicolas. Shape derivative of the Cheeger constant. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 348-358. doi: 10.1051/cocv/2014018

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