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Extremal functions for Morrey's inequality in convex domains
MIT, Dept Math, Cambridge, MA 02139 USA.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2019 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 375, no 3-4, p. 1721-1743Article in journal (Refereed) Published
Abstract [en]

For a bounded domain Omega subset of R-n and p > n, Morrey's inequality implies that there is c > 0 such that c parallel to u parallel to(p)(infinity) <= integral(Omega) vertical bar Du vertical bar(p) dx for each u belonging to the Sobolev space W-0(1,p) (Omega). We show that the ratio of any two extremal functions is constant provided that Omega is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian.

Place, publisher, year, edition, pages
2019. Vol. 375, no 3-4, p. 1721-1743
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-397123DOI: 10.1007/s00208-018-1775-8ISI: 000492595100023OAI: oai:DiVA.org:uu-397123DiVA, id: diva2:1374253
Funder
Swedish Research Council, 2012-3124Swedish Research Council, 2017-03736Available from: 2019-11-29 Created: 2019-11-29 Last updated: 2019-11-29Bibliographically approved

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