Optimal partition problems for the fractional laplacian

Antonella Ritorto

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arXiv · 1703.05642

Optimal partition problems for the fractional laplacian

Antonella Ritorto

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Abstract

In this work, we prove an existence result for an optimal partition problem of the form $$\min \{F_s(A_1,\dots,A_m)\colon A_i \in \mathcal{A}_s, \, A_i\cap A_j =\emptyset \mbox{ for } i\neq j\},$$ where $F_s$ is a cost functional with suitable assumptions of monotonicity and lowersemicontinuity, $\mathcal{A}_s$ is the class of admissible domains and the condition $A_i\cap A_j =\emptyset$ is understood in the sense of the Gagliardo $s$-capacity, where $0<s<1$. Examples of this type of problem are related to the fractional eigenvalues. In addition, we prove some type of convergence of the $s$-minimizers to the minimizer of the problem with $s=1$, studied in \cite{Bucur-Buttazzo-Henrot}.

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Antonella Ritorto. 2017-03-16. Optimal partition problems for the fractional laplacian. https://arxiv.org/abs/1703.05642

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