Anisotropic fractional perimeters

Abstract

The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\rn$ is shown to converge to the anisotropic perimeter with respect to the moment body of $K$ as $s\to 1^{-}$. A corresponding result is established for anisotropic fractional $s$-seminorms on $BV(\rn)$ (generalizing results of Bourgain, Brezis & Mironescu and Dávila). The minimizers of the anisotropic fractional $s$-isoperimetric inequality with respect to $K$ are shown to converge to the moment body of $K$. Anisotropic fractional Sobolev inequalities are established.

References in corpus (1)

• Gamma-convergence of nonlocal perimeter functionals

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• Affine fractional $L^p$ Sobolev inequalities
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• Existence and instability of deformed catenoidal solutions for fractional Allen--Cahn equation
• Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media
• Regularized Riesz energies of submanifolds