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A general class of free boundary problems for fully nonlinear parabolic equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2015 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 194, no 4, 1123-1134 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we consider the fully nonlinear parabolic free boundary problem { F(D(2)u) - partial derivative(t)u = 1 a.e. in Q(1) boolean AND Omega vertical bar D(2)u vertical bar + vertical bar partial derivative(t)u vertical bar <= K a.e. in Q(1)\Omega, where K > 0 is a positive constant, and Omega is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W-x(2,) (n) boolean AND W-t(1,) (n) solutions are locally C-x(1,) (1) boolean AND C-t(0,) (1) inside Q(1). A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result in Caffarelli and Huang (Duke Math J 118(1): 1-17, 2003). Once optimal regularity for u is obtained, we also show regularity for the free boundary partial derivative Omega boolean AND Q(1) under the extra condition that Omega superset of{u not equal 0}, and a uniform thickness assumption on the coincidence set {u = 0}.

Place, publisher, year, edition, pages
2015. Vol. 194, no 4, 1123-1134 p.
Keyword [en]
Free boundaries, Regularity, Parabolic fully nonlinear
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-173446DOI: 10.1007/s10231-014-0413-7ISI: 000359804400009Scopus ID: 2-s2.0-84931568511OAI: oai:DiVA.org:kth-173446DiVA: diva2:853775
Note

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved

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Shahgholian, Henrik

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