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A parabolic two-phase obstacle-like equation
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2009 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 221, no 3, 861-881 p.Article in journal (Refereed) Published
Abstract [en]

For the parabolic obstacle-problem-like equation Delta u - partial derivative(t)u = lambda(+chi{u>0})-lambda(-chi{u<0}), where lambda(+) and lambda(-) are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary delta {u > 0} boolean OR partial derivative{u < 0} is in a neighborhood of each "branch point" the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper [Henrik Shahgholian, Nina Uraltseva, Georg S. Weiss, The two-phase membrane problem-regularity in higher dimensions, Int. Math. Res. Not. (8) (2007)] to the parabolic case. There are substantial difficulties in the parabolic case clue to the fact that the time derivative of the solution is in general not a continuous function. Our result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.

Place, publisher, year, edition, pages
2009. Vol. 221, no 3, 861-881 p.
Keyword [en]
Free boundary, Singular point, Branch point, Membrane, Obstacle, problem, Regularity, Global solution, Blow-up, free-boundary, regularity
URN: urn:nbn:se:kth:diva-18369DOI: 10.1016/j.aim.2009.01.011ISI: 000265616000006ScopusID: 2-s2.0-64249143572OAI: diva2:336415
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2011-01-14Bibliographically approved

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