Anisotropic Fractional Sobolev Norms

Abstract

Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^{-}$. The anisotropic $s$-seminorms of $f$ defined by a norm on $\rn$ with unit ball $K$ are shown to converge to the anisotropic Sobolev seminorm of $f$ defined by the norm with unit ball $\,\ompd K$, the polar $L_p$ moment body of $K$. The limiting behavior for $s\to 0^{+}$ is also determined (extending results by Maz${}^{\prime }$ya & Shaposhnikova).

References in corpus (4)

• General Lp affine isoperimetric inequalities
• Anisotropic fractional perimeters
• SL(n)-Contravariant $L_p$-Minkowski Valuations
• SL(n)-Covariant $L_p$-Minkowski Valuations

Cited by in corpus (11)

• A Remake of Bourgain-Brezis-Mironescu Characterization of Sobolev Spaces
• Anisotropic Sobolev Capacity with Fractional Order
• Asymptotic behaviours in Fractional Orlicz-Sobolev spaces on Carnot groups
• Bourgain-Brezis-Mironescu formula for $W_q^{s,p}$-spaces in arbitrary domains
• Affine fractional $L^p$ Sobolev inequalities
• Asymmetric anisotropic fractional Sobolev norms
• On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited
• On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity
• Optimal geometric estimates for fractional Sobolev capacities
• An affine Orlicz Polya-Szego principle
• On the $\Gamma$-limit of weighted fractional energies