Boundary behavior of positive solutions of the heat equation on a semi-infinite slab
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- by B. A. Mair PDF
- Trans. Amer. Math. Soc. 295 (1986), 687-697 Request permission
Abstract:
In this paper, the abstract Fatou-Naim-Doob theorem is used to investigate the boundary behavior of positive solutions of the heat equation on the semi-infinite slab $X = {{\mathbf {R}}^{n - 1}} \times {{\mathbf {R}}_ + } \times (0,T)$. The concept of semifine limit is introduced, and relationships are obtained between fine, semifine, parabolic, one-sided parabolic and two-sided parabolic limits at points on the parabolic boundary of $X$. A Carleson-Calderón-type local Fatou theorem is also obtained for solutions on a union of two-sided parabolic regions.References
- Heinz Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics, No. 22, Springer-Verlag, Berlin-New York, 1966 (German). Ausarbeitung einer im Sommersemester 1965 an der Universität Hamburg gehaltenen Vorlesung. MR 0210916
- M. Brelot and J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) 13 (1963), no. fasc. 2, 395–415 (French). MR 196107
- A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 68 (1950), 47–54. MR 32863, DOI 10.1090/S0002-9947-1950-0032863-9
- Lennart Carleson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat. 4 (1962), 393–399 (1962). MR 159013, DOI 10.1007/BF02591620 E. B. Fabes, N. Garaofalo and S. Salsa, A backward Harnack inequality and Fatou theorem for non-negative solutions of parabolic equations, Univ. of Minnesota Math. Rep. 83-117.
- B. Frank Jones Jr. and C. C. Tu, Non-tangential limits for a solution of the heat equation in a two-dimensional $\textrm {Lip}_{\alpha }$ region, Duke Math. J. 37 (1970), 243–254. MR 259388
- Robert Kaufman and Jang Mei Wu, Parabolic potential theory, J. Differential Equations 43 (1982), no. 2, 204–234. MR 647063, DOI 10.1016/0022-0396(82)90091-2
- John T. Kemper, Temperatures in several variables: Kernel functions, representations, and parabolic boundary values, Trans. Amer. Math. Soc. 167 (1972), 243–262. MR 294903, DOI 10.1090/S0002-9947-1972-0294903-6
- Adam Korányi and J. C. Taylor, Fine convergence and parabolic convergence for the Helmholtz equation and the heat equation, Illinois J. Math. 27 (1983), no. 1, 77–93. MR 684542
- B. A. Mair, Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab, Trans. Amer. Math. Soc. 284 (1984), no. 2, 583–599. MR 743734, DOI 10.1090/S0002-9947-1984-0743734-7
- Bernard Mair and J. C. Taylor, Integral representation of positive solutions of the heat equation, Théorie du potentiel (Orsay, 1983) Lecture Notes in Math., vol. 1096, Springer, Berlin, 1984, pp. 419–433. MR 890370, DOI 10.1007/BFb0100123
- Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
- J. C. Taylor, An elementary proof of the theorem of Fatou-Naïm-Doob, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 153–163. MR 670103
- C. C. Tu, Non-tangential limits of a solution of a boundary-value problem for the heat equation, Math. Systems Theory 3 (1969), 130–138. MR 249840, DOI 10.1007/BF01746519
- D. V. Widder, The heat equation, Pure and Applied Mathematics, Vol. 67, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0466967
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 295 (1986), 687-697
- MSC: Primary 35K20; Secondary 31B25
- DOI: https://doi.org/10.1090/S0002-9947-1986-0833703-2
- MathSciNet review: 833703