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arXiv · 1304.0703

Anisotropic Fractional Sobolev Norms

Monika Ludwig
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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$'$ya & Shaposhnikova).

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Monika Ludwig. 2013-04-02. Anisotropic Fractional Sobolev Norms. https://doi.org/10.1016/j.aim.2013.10.024

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