The fractional Cheeger problem

By Lorenzo Brasco, Erik Lindgren, Enea Parini. 2013

210 citations recorded by the source corpus.

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a8bea51b6b9172088f106d1e2250252734ef12d

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The second eigenvalue of the fractional p-Laplacian (2014, background): “Lindgren and Lindqvist in [21] and investigated by several authors afterwards,we cite for example [4, 13, 17].”
Fractional p-Laplacian evolution equations (2016, background): “We refer to [19] where one can find a description of most of the useful properties of the fractional Sobolev spaces (see also [12]).”
Stability of variational eigenvalues for the fractional p−Laplacian (2015, methodology): “Since Ω is Lipschitz, the latter coincides with the space used in [6, 7] and defined as the completion of C∞ 0 (Ω) with respect to [ · ]W s,p(RN ).”
Convexity properties of Dirichlet integrals and Picone-type inequalities (2014, background): “the reader is referred to [8, 19, 22] for a more detailed account about the case q = p.”
Nonlocal Harnack inequalities (2014, background): “Also, we would like to mention the analysis in the papers [2, 9] where some basic results for fractional p-eigenvalues have been proven.”
A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators (2021, methodology): “The optimal shape problems for the first eigenvalue of the fractional Laplacian with homogeneous external datum was addressed in [3, 13, 56, 11], showing that the ball is the optimizer.”
Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems (2021, background): “1 nonlocal case, see Di Nezza-Palatucci-Valdinoci [34], Lindgren-Lindqvist [54], Brasco-LindgrenParini [18], Brasco-Parini [19], Franzina-Palatucci [44], Nguyen-Dı́az-Quoc-Hung [58] and the references therein.”