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A free boundary problem with log term singularity
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-1316-7913
2017 (English)In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 19, no 3, p. 351-369Article in journal (Refereed) Published
Abstract [en]

We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar del v vertical bar(2)/2 - v(+)(log v - 1))dx -> min which should be minimized in some natural admissible class of non-negative functions. Here, v(+) = max{0, v}. The Euler-Lagrange equation associated with (sic) is -Delta u = chi({u>0}) log u, which becomes singular along the free boundary partial derivative{u > O}. Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like r(2)vertical bar log r vertical bar. This estimate is crucial in the study of analytic and geometric properties of the free boundary.

Place, publisher, year, edition, pages
2017. Vol. 19, no 3, p. 351-369
Keywords [en]
Free boundary, regularity theory, logarithmic singularity, porosity
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-220303DOI: 10.4171/IFB/385ISI: 000416727600002Scopus ID: 2-s2.0-85031497391OAI: oai:DiVA.org:kth-220303DiVA, id: diva2:1168842
Funder
Swedish Research Council
Note

QC 20171221

Available from: 2017-12-21 Created: 2017-12-21 Last updated: 2017-12-21Bibliographically approved

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Shahgholian, Henrik

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