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Shape derivative of the Cheeger constant. (English) Zbl 1315.49018

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

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting
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References:

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