This paper deals with the existence of the shape derivative of the Cheeger constant h1(Ω) of a bounded domain Ω. We prove that if Ω admits a unique Cheeger set, then the shape derivative of h1(Ω) exists, and we provide an explicit formula. A counter-example shows that the shape derivative may not exist without the uniqueness assumption.
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Parini et al. (2014) studied this question.
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