Existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters (Q515931)

In the present paper the authors establish basic existence and partial regularity results for the fractional isoperimetric problem with multiple volume constraints. If \(E\subset \mathbb{R}^n, \;n\geq 2\), and \(s\in (0,1)\), then the fractional perimeter of order \(s\) of \(E\) is defined as \(P_s(E)=\int_{\mathbb{R}^n}w_E(x)dx =\int_Edx\int_{E^c}dy/|x-y|^{n+s}\). First, the authors start the study, in the fractional setting, of another classical geometric variational problem, namely the isoperimetric problem with multiple volume constraints. The main result is the following Theorem 1.1. For every \(m\in \mathbb{R}^n_+\) there exists an isoperimetric cluster \(\mathcal{E}\) with \(m(\mathcal{E})=m\). If we set \(\partial \mathcal{E}=\{x\in \mathbb{R}^n: \exists h=1,\dots, N\), such that \(0<|\mathcal{E}(h)\cap B_r(x)|<|B_r(x)|, \forall r>0\}\), then \(\partial \mathcal{E}\) is bounded and there exists a closed set \(\Sigma (\mathcal{E})\subset \partial \mathcal{E}\) of dimension less than or equal to \(n-2\) (namely, such that \(\mathcal{H}^{n-2+\varepsilon}(\Sigma (\mathcal{E}))=0\) for every \(\varepsilon>0\)) if \(n\geq 3, \Sigma(\mathcal{E})\) is discrete if \(n=2\), and \(\partial \mathcal{E}\setminus \Sigma (\mathcal{E})\) is a \(C^{1,\alpha}\)-hypersurface in \(\mathbb{R}^n\) for some \(\alpha \in (0,1)\). The present paper naturally opens two kinds of questions, which are closely related: first, understanding singularities of fractional isoperimetric clusters and, second, characterizing fractional isoperimetric clusters in some basic cases. The paper is divided into two sections. In Section 2 , the authors prove the existence part in Theorem 1.1 by adapting to the fractional setting Almgren's original proof. In Section 3 the authors prove the partial regularity assertion in Theorem 1.1. Just like in the case of fractional perimeter minimizing boundaries, they exploit an extension problem to obtain a monotonicity formula, showing that nearby most points of the boundary only two chambers of the isoperimetric cluster are present. When this happens, they show that the neighboring chambers locally almost-minimize fractional perimeter, and they apply a known result from \textit{M. C. Caputo} and \textit{N. Guillen} [``Regularity for non-local almost minimal boundaries and applications'', Preprint, \url{arXiv:1003.2470}], to prove the \(C^{1,\alpha}\)-regularity.