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Regularity for non-local almost minimal boundaries and applications

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arxiv 1003.2470 v2 pith:GELMMWUC submitted 2010-03-12 math.AP math.DG

M. Cristina Caputo , Nestor Guillen This is my paper · ORCID
classification math.APmath.DG
keywords non-localminimalregularityalmostboundariesapplicationstheoryalmgren
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abstract

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-De Giorgi-Tamanini regularity theory. The main result has several applications, among these $C^{1,\alpha}$ regularity for sets with prescribed non-local mean curvature in $L^p$ and regularity of solutions to non-local obstacle problems.

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    Limits of vectorial fractional Allen-Cahn solutions are stationary nonlocal minimal partitions, and minimizing 3-partitions are smooth outside a small singular set, even under a reversed triangle inequality for s clos...

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