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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab


Author: B. A. Mair
Journal: Trans. Amer. Math. Soc. 284 (1984), 583-599
MSC: Primary 35K10; Secondary 31B25
DOI: https://doi.org/10.1090/S0002-9947-1984-0743734-7
Erratum: Trans. Amer. Math. Soc. 291 (1985), 381.
MathSciNet review: 743734
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Abstract: This paper investigates the boundary behaviour of positive solutions of the equation $ Lu = 0$, where $ L$ is a uniformly parabolic second-order differential operator in divergence form having Hölder-continuous coefficients on $ X = {{\mathbf{R}}^n} \times (0,T)$, where $ 0 < T < \infty $. In particular, the notion of semithinness for the potential theory on $ X$ associated with $ L$ is introduced, and the relationships between fine, semifine and parabolic convergence at points of $ {{\mathbf{R}}^n} \times \{ 0 \}$ are studied.

The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of $ Lu = 0$ on $ X$ has parabolic limits Lebesgue-almost-everywhere on $ {{\mathbf{R}}^n} \times \{ 0 \}$. Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0743734-7
Keywords: Harmonic, fine limit, semifine limit, parabolic limit, local Fatou theorem
Article copyright: © Copyright 1984 American Mathematical Society

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