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A nonlocal isoperimetric problem with density perimeter. (English) Zbl 1455.49028

Summary: We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent \(\alpha\), under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter \(\gamma\). We show that for a wide class of density functions the energy admits a minimizer for any value of \(\gamma\). Moreover these minimizers are bounded. For monomial densities of the form \(|x|^p\) we prove that when \(\gamma\) is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the \(\gamma \rightarrow 0\) limit corresponds, under a suitable rescaling, to a small mass \(m=|\Omega |\rightarrow 0\) limit when \(p<d-\alpha +1\), but to a large mass \(m\rightarrow \infty\) for powers \(p>d-\alpha +1\).

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting
49J10 Existence theories for free problems in two or more independent variables
28A75 Length, area, volume, other geometric measure theory
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