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A Fatou theorem for the equation $u_t=\Delta (u-1)_+$

Author: Marianne K. Korten
Journal: Proc. Amer. Math. Soc. 128 (2000), 439-444
MSC (1991): Primary 35K65, 31A20
Published electronically: September 24, 1999
MathSciNet review: 1670395
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Abstract: In one space dimension and for a given function $ u_I (x) \in C_0 ^ {\infty}$ (say such that $ u_I (x) > 1$ in some interval), the equation $ u_t = \Delta (u-1)_+$ can be thought of as describing the energy per unit volume in a Stefan-type problem where the latent heat of the phase change is given by $ 1-u_I (x) $. Given a solution $ 0 \leq u \in L^1 _{\mathrm{loc}} (\mathbb{R} ^n \times (0,T)) $ to this equation, we prove that for a.e. $x_0 \in \mathbb{R} ^n $, there exists $ \lim _{(x,t) \in \Gamma _{\beta} ^k (x_0),\; (x,t) \to x_0} (u(x,t) -1)_+ =( f(x_0)-1)_+,$ where $f =\partial \mu / \partial |\;|$ is the Radon-Nikodym derivative of the initial trace $\mu$ with respect to Lebesgue measure and $\Gamma _{\beta} ^k (x_0) = \{ (x,t): |x-x_0| < \beta \sqrt t,\; 0<t<k \}$ are the parabolic ``non-tangential" approach regions. Since only $(u-1)_+$ is continuous, while $u$ is usually not, $ \lim _{(x,t) \in \Gamma _{\beta} ^k (x_0),\; (x,t) \to x_0} u(x,t) = f(x_0) $ does not hold in general.

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Marianne K. Korten
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pab. No. 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina; Instituto Argentino de Matemática (CONICET), Saavedra 15, 3er. piso, 1083 Buenos Aires, Argentina
Address at time of publication: Department of Mathematics, University of Liousville, Louisville, Kentucky 40292

Received by editor(s): February 28, 1998
Published electronically: September 24, 1999
Additional Notes: This research was partially supported by PIDs 3668/92 and 3164/92-CONICET and EX 071-UBA
Dedicated: Dedicated to the memory of Eugene Fabes
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society