## 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

### Keywords:

shape derivative; Cheeger constant; 1-Laplacian
Full Text:

### References:

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