Pointwise regularity of the free boundary for the parabolic obstacle problem
نویسندگان
چکیده
We study the parabolic obstacle problem ∆u− ut = fχ{u>0}, u ≥ 0, f ∈ L 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) C1-surface and that the set of singular points is locally contained in a union of (parabolic) C1 manifolds. AMS Classification: 35R35.
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