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Transactions of the American Mathematical Society

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Behavior near the boundary
of positive solutions
of second order parabolic equations. II


Authors: E. B. Fabes, M. V. Safonov and Yu Yuan
Journal: Trans. Amer. Math. Soc. 351 (1999), 4947-4961
MSC (1991): Primary 35K10, 35B05; Secondary 35B45, 31B25
DOI: https://doi.org/10.1090/S0002-9947-99-02487-3
Published electronically: August 10, 1999
MathSciNet review: 1665328
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Abstract | References | Similar Articles | Additional Information

Abstract: A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder $Q=\Omega \times \left(0,T\right)$ which vanish on $\partial _xQ=\partial\Omega\times \left(0,T\right)$, where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$. This inequality is applied to the proof of the Hölder continuity of the quotient of two positive solutions vanishing on a portion of $\partial _xQ.$


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  • [AC] I. Athanasopoulos and L. A. Caffarelli, A theorem of real analysis and its application to free boundary problems, Comm. Pure and Appl. Math., 38(1985), 499-502. MR 86j:49062
  • [ACS] I. Athanasopoulos, L. A. Caffarelli and S. Salsa, Caloric functions in Lipschitz domains and regularity of solutions to phase transition problems, Ann. Math., 143(1996), 413-434. MR 97e:35074
  • [B] P. E. Bauman, Positive solutions of elliptic equations in non-divergence form and their adjoints, Arkiv fur Mathematik, 22(1984), 153-173. MR 86m:35008
  • [CFMS] L. A. Caffarelli, E. B. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana J. of Math., 30(1981), 621-640. MR 83c:35040
  • [FGMS] E. B. Fabes, N. Garofalo, S. Marín-Malave and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Revista Math. Iberoamericana, 4(1988), 227-251. MR 91e:35092
  • [FGS] E. B. Fabes, N. Garofalo and S. Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. of Math., 30(1986), 536-565. MR 88d:35089
  • [FK] E. B. Fabes and C. E. Kenig, Examples of singular parabolic measures and singular transition probability densities, Duke Math. J., 48(1981), 845-856. MR 86j:35081
  • [FS] E. B. Fabes and M. V. Safonov, Behavior near boundary of positive solutions of second order parabolic equations, J. Fourier Anal. and Appl., Special Issue: Proceedings of the Conference El Escorial 96, 3(1997), 871-882. MR 99d:35071
  • [FSt] E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old idea of Nash, Arch. Rational Mech. Anal., 96(1986), 327-338. MR 88b:35037
  • [G] N. Garofalo, Second order parabolic equations in nonvariational form: boundary Harnack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura Appl., 138(1984), 267-296. MR 87f:35115
  • [JK] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46(1982), 80-147. MR 84d:31005b
  • [K] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Nauka, Moscow, 1985 in Russian; English transl.: Reidel, Dordrecht, 1987. MR 88d:35005
  • [KS] N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Izvestia Akad. Nauk SSSR, ser. Matem., 44(1980), 161-175 in Russian; English transl. in Math. USSR Izvestija, 16(1981), 151-164. MR 83c:35059
  • [LSU] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva , Linear and quasi-linear equations of parabolic type, Nauka, Moscow, 1967 in Russian; English transl.: Amer. Math. Soc., Providence, RI, 1968. MR 39:3159a
  • [M1] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure and Appl. Math., 17(1964), 101-134; and correction in: Comm. Pure and Appl. Math., 20(1967), 231-236. MR 34:6288
  • [M2] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure and Appl. Math., 24(1971), 727-740. MR 44:5603
  • [S] M. V. Safonov, Abstracts of Communications, Third Vilnius conference on probability theory and mathematical statistics, June 22-27, 1981.
  • [SY] M. V. Safonov and Yu Yuan, Doubling properties for second order parabolic equations, to appear in Ann. Math.
  • [Sl] S. Salsa, Some properties of nonnegative solutions of parabolic differential operators, Ann. Mat. Pura Appl., 128(1981), 193-206. MR 83j:35078
  • [T] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure and Appl. Math., 21(1968), 205-226.

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Additional Information

M. V. Safonov
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455-0100
Email: safonov@math.umn.edu

Yu Yuan
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455-0100
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: yyuan@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02487-3
Keywords: Harnack inequality, H\"{o}lder continuity, caloric measure.
Received by editor(s): August 4, 1997
Published electronically: August 10, 1999
Additional Notes: The second and third authors are partially supported by NSF Grant No. DMS-9623287
Article copyright: © Copyright 1999 American Mathematical Society

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