Remote Access Proceedings of the American Mathematical Society
Green Open Access

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

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?)

  • [AK] D. Andreucci, M.K. Korten, Initial traces of solutions to a one-phase Stefan problem in an infinite strip, Rev. Mat. Iberoamericana, Vol. 9, No. 2 (1993), 315-332. MR 94m:35319
  • [B] J. E. Bouillet, Signed solutions to diffusion-heat conduction equations, Free Boundary Problems: Theory and Applications, Proc. Int. Colloq. Irsee/Ger. 1987, Vol. II, Pitman Res. Notes Math. Ser. 186 (1990), 480-485.
  • [BKM] J. E. Bouillet, M. K. Korten and V. Márquez, Singular limits and the ``Mesa" problem, Rev. Union Mat. Argentina, Vol. 41 (1998), no. 1, 27-40.
  • [C] A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 68 (1950), 47-54. MR 11:357e
  • [DFK] B. E. J. Dahlberg, E. Fabes and C. E. Kenig, A Fatou theorem for solutions of the porous medium equation, Proc. Amer. Math. Soc. 91 (1984), 205-212. MR 85e:35064
  • [DB] E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. (4), CXXX (1982), 131-176. MR 83k:35045
  • [H] K. M. Hui, Fatou theorem for the solutions of some nonlinear equations, Jl. Math. Anal. Applic. 183 (1994), 37-52. MR 95c:35125
  • [K] M. K. Korten, Non-negative solutions of $u_t = \Delta (u-1)_+$: Regularity and uniqueness for the Cauchy problem, Nonl. Anal., Th., Meth. and Appl, Vol. 27, No. 5 (1996), 589-603. MR 97h:35089

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

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

American Mathematical Society