Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35K65, 31A20

Retrieve articles in all journals with MSC (1991): 35K65, 31A20


Additional Information

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
Email: mkorten@dm.uba.ar, korten@louisville.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05386-1
PII: S 0002-9939(99)05386-1
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