The Mathematical Institute, University of Oxford, Eprints Archive

On the maximal Sobolev regularity
of distributions supported by subsets of Euclidean space

Hewett, D. P. and Moiola, A (2015) On the maximal Sobolev regularity
of distributions supported by subsets of Euclidean space.
Technical Report. Unspecified. (Submitted)

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Abstract

Given a subset $E$ of $\R^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\R$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p (\R^n)$ that is supported in $E$? For sets of zero Lebesgue measure we show, using results on certain set capacities from classical potential theory, that the maximal regularity is non-positive, and is characterised by the Hausdorff dimension of $E$, improving known results. We classify all possible maximal regularities, as functions of $p$, together with the sets of values of $p$ for which the maximal regularity is attained, and construct concrete examples for each case. For sets with positive measure the maximal regularity is non-negative, but appears more difficult to characterise in terms of geometrical properties of $E$. We present some partial results relating to the latter case, namely lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterising the regularity that can be achieved on certain special classes of sets, such as $d$-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations. [brace not closed]

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1902
Deposited By: Helen Taylor
Deposited On:16 Sep 2015 06:35
Last Modified:16 Sep 2015 06:35

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