A Fatou theorem for the equation

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 | References | Similar Articles | Additional Information

Abstract: In one space dimension and for a given function (say such that in some interval), the equation 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 . Given a solution to this equation, we prove that for a.e. , there exists where is the Radon-Nikodym derivative of the initial trace with respect to Lebesgue measure and are the parabolic ``non-tangential" approach regions. Since only is continuous, while is usually not, does not hold in general.

**[AK]**D. Andreucci and M. K. Korten,*Initial traces of solutions to a one-phase Stefan problem in an infinite strip*, Rev. Mat. Iberoamericana**9**(1993), no. 2, 315–332. MR**1232846**, 10.4171/RMI/139**[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**0032863**, 10.1090/S0002-9947-1950-0032863-9**[DFK]**Björn E. J. Dahlberg, Eugene B. Fabes, and Carlos E. Kenig,*A Fatou theorem for solutions of the porous medium equation*, Proc. Amer. Math. Soc.**91**(1984), no. 2, 205–212. MR**740172**, 10.1090/S0002-9939-1984-0740172-3**[DB]**Emmanuele DiBenedetto,*Continuity of weak solutions to certain singular parabolic equations*, Ann. Mat. Pura Appl. (4)**130**(1982), 131–176 (English, with Italian summary). MR**663969**, 10.1007/BF01761493**[H]**Kin Ming Hui,*Fatou theorem for the solutions of some nonlinear equations*, J. Math. Anal. Appl.**183**(1994), no. 1, 37–52. MR**1273430**, 10.1006/jmaa.1994.1129**[K]**Marianne K. Korten,*Nonnegative solutions of 𝑢_{𝑡}=Δ(𝑢-1)₊: regularity and uniqueness for the Cauchy problem*, Nonlinear Anal.**27**(1996), no. 5, 589–603. MR**1396031**, 10.1016/0362-546X(95)00137-K

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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:
https://doi.org/10.1090/S0002-9939-99-05386-1

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