Journal
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Volume 54, Issue 1, Pages 299-347Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00526-014-0787-9
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Funding
- ANR project MICA [ANR-08-BLAN-0082]
- ANR project HJnet [ANR-12-BS01-0008-01]
- Chair Mathematical modelling and numerical simulation, F-EADS-Ecole Polytechnique-INRIA
- Royal Swedish Academy of Sciences
- NTNU
- MSRI
- Swedish Research Council [2012-3124]
- European Research Council under the European Union [321186]
- Agence Nationale de la Recherche (ANR) [ANR-08-BLAN-0082] Funding Source: Agence Nationale de la Recherche (ANR)
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We study the parabolic obstacle problem Delta u - u(t) = f chi((u>0)), u >= 0, f is an element of L-p with f(0) = 1 and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that f is Dini continuous, we prove that the set of regular points is locally a (parabolic) C-1-surface and that the set of singular points is locally contained in a union of (parabolic) C-1 manifolds.
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