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

Hewett, D. and Moiola, A. (2017) On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space. Analysis and Applications, 15 (5). pp. 731-770. ISSN 1793-6861

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To link to this item DOI: 10.1142/S021953051650024X

Abstract/Summary

This paper concerns the following question: given a subset E of Rn with empty interior and an integrability parameter 1<p<infinity, what is the maximal regularity s in R for which there exists a non-zero distribution in the Bessel potential Sobolev space Hs,p(Rn) that is supported in E? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of E, sharpening previous results. Furthermore, we provide a full classification of 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. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new 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.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:68458
Uncontrolled Keywords:Bessel potential Sobolev spaces; (s, p)-nullity; polar set; set of uniqueness; capacity; Hausdorff dimension; Cantor sets.
Publisher:World Scientific

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