Shape derivative of the Cheeger constant

Parini, Enea; Saintier, Nicolas

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 348-358

This paper deals with the existence of the shape derivative of the Cheeger constant $h_1\left(\Omega \right)$ of a bounded domain $\Omega$. We prove that if $\Omega$ admits a unique Cheeger set, then the shape derivative of $h_1\left(\Omega \right)$ exists, and we provide an explicit formula. A counter-example shows that the shape derivative may not exist without the uniqueness assumption.

Classification : 49Q10, 49Q20
Keywords: Shape derivative, CHEEGER constant, 1-Laplacian

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