Near-sphere lattices with constant nonlocal mean curvature
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We are concerned with unbounded sets of RN whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension M=N. Our CNMC sets form a C2 branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point. Thus, the new translated near-balls (or near-spheres) appear from infinity. We find their exact asymptotic shape as the parameter tends to infinity.
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CitationCabre, X.; Fall, M.; Weth, T. Near-sphere lattices with constant nonlocal mean curvature. "Mathematische Annalen", Abril 2018, vol. 370, núm. 3-4, p. 1513-1569.