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Wiener's criterion for parabolic equations with variable coefficients and its consequences

Authors: Nicola Garofalo and Ermanno Lanconelli
Journal: Trans. Amer. Math. Soc. 308 (1988), 811-836
MSC: Primary 35K20; Secondary 31B10, 31B20
MathSciNet review: 951629
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Abstract: In a bounded set in $ {{\mathbf{R}}^{n + 1}}$ we study the problem of the regularity of boundary points for the Dirichlet problem for a parabolic operator with smooth coefficients. We give a geometric characterization, modelled on Wiener's criterion for Laplace's equation, of those boundary points that are regular. We also present some important consequences. Here is the main one: a point is regular for a variable coefficient operator if and only if it is regular for the constant coefficient operator obtained by freezing the coefficients at that point.

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  • [A] D. G. Aronson, Non negative solutions of linear parabolic equations, Ann. Norm. Sup. Pisa 22 (1968), 607-694. MR 0435594 (55:8553)
  • [Ba] H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math., vol. 22, Springer-Verlag, 1966. MR 0210916 (35:1801)
  • [BGM] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemanniene, Lecture Notes in Math., vol. 194, Springer-Verlag, 1971. MR 0282313 (43:8025)
  • [B] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., vol. 175, Springer-Verlag, 1971. MR 0281940 (43:7654)
  • [EG] L. C. Evans and R. F. Gariepy, Wiener's criterion for the heat equation, Arch. Rational Mech. Anal. 78 (1982), 293-314. MR 653544 (83g:35047)
  • [FG] E. B. Fabes and N. Garofalo, Mean value properties of solutions to parabolic equations with variable coefficients, J. Math. Anal. Appl. 121 (1987), 305-316. MR 872228 (88b:35088)
  • [Fe] H. Federer, Geometric measure theory, Springer-Verlag, 1969.
  • [Fr] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 0181836 (31:6062)
  • [Fu] W. Fulks, A mean value theorem for the heat equation, Proc. Amer. Math. Soc. 17 (1966), 6-11. MR 0192200 (33:427)
  • [GL] N. Garofalo and E. Lanconelli, Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients, preprint. MR 980595 (90i:35040)
  • [GZ] R. F. Gariepy and W. P. Ziemer, Thermal capacity and boundary regularity, J. Differential Equations 45 (1982), 374-388. MR 672714 (84c:35062)
  • [K] Y. Kannai, Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. 2 (1977), 781-830. MR 0603299 (58:29247)
  • [L1] E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. 97 (1973), 83-114. MR 0372226 (51:8442)
  • [L2] -, Sul problema di Dirichlet per equazioni paraboliche del secondo ordine a coefficienti discontinue, Ann. Mat. Pura Appl. 106 (1975), 11-37. MR 0399659 (53:3502)
  • [L3] -, Sul confronto della regolarità dei punti di frontiera rispetto ad operatori lineari parabolici diversi, Ann. Mat. Pura Appl. 114 (1977), 207-227. MR 0481527 (58:1643)
  • [La] E. M. Landis, Necessary and sufficient conditions for regularity of a boundary point in the Dirichlet problem for the heat-conduction equation, Soviet Math. 10 (1969), 380-384.
  • [LSW] W. Littman, G. Stampacchia and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa 17 (1963), 45-79. MR 0161019 (28:4228)
  • [N] A. A. Nohruzov, On some regularity criteria for boundary points for linear and quasi-linear parabolic equations, Dokl. Akad. Nauk SSSR 209 (1973). (Russian)
  • [Pe] I. Petrovsky, Zur ersten Randwertaufgabe der Warmeleitungsgleichung, Compositio Math. 1 (1935), 383-419. MR 1556900
  • [P1] B. Pini, Sulle equazioni a derivate parziali lineari del secondo ordine in due variabili di tipo parabolico, Ann. Mat. Pura Appl. 32 (1951), 179-204. MR 0046541 (13:750e)
  • [P2] -, Maggioranti e minoranti delle soluzioni delle equazioni paraboliche, Ann. Mat. Pura Appl. 37 (1954), 249-264. MR 0066540 (16:593f)
  • [P3] -, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova 23 (1954), 422-434. MR 0065794 (16:485c)
  • [Wa1] N. A. Watson, A theory of temperatures in several variables, Proc. London Math. Soc. (3) 26 (1973), 385-417. MR 0315289 (47:3838)
  • [Wa2] -, Thermal capacity, Proc. London Math. Soc. 37 (1978), 342-362. MR 507610 (80b:31005)
  • [W] N. Wiener, The Dirichlet problem, J. Math. Phys. 3 (1924), 127-146.
  • [Z] W. P. Ziemer, Behavior at the boundary of solutions of quasi-linear parabolic equations, J. Differential Equations 35 (1980), 291-305. MR 563383 (81c:35073)

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Keywords: Parabolic equations, Wiener's criterion, mean value properties
Article copyright: © Copyright 1988 American Mathematical Society

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