Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on Krylov-Tso's parabolic inequality

Author: Luis Escauriaza
Journal: Proc. Amer. Math. Soc. 115 (1992), 1053-1056
MSC: Primary 35K20; Secondary 35B05
MathSciNet review: 1092918
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if $ u$ is a solution to $ \sum\nolimits_{i,j = 1}^n {{a_{ij}}(x,t){D_{ij}}u(x,t) - {D_t}u(x,t) = \phi (x)} $ on a cylinder $ {\Omega _T} = \Omega \times (0,T)$, where $ \Omega $ is a bounded open set in $ {{\mathbf{R}}^n},T > 0$, and $ u$ vanishes continuously on the parabolic boundary of $ {\Omega _T}$. Then the maximum of $ u$ on the cylinder is bounded by a constant $ C$ depending on the ellipticity of the coefficient matrix $ ({a_{ij}}(x,t))$, the diameter of $ \Omega $, and the dimension $ n$ times the $ {L^n}$ norm of $ \phi $ in $ \Omega $.

References [Enhancements On Off] (What's this?)

  • [1] A. D. Aleksandrov, Majorants of solutions of linear equations of order two, Vestnik Leningrad. Univ. 21 (1966), no. 1, 5–25 (Russian, with English summary). MR 0199540
  • [2] I. Ja. Bakel′man, On the theory of quasilinear elliptic equations, Sibirsk. Mat. Ž. 2 (1961), 179–186 (Russian). MR 0126604
  • [3] Bartolomé Barceló, Luis Escauriaza, and Eugene Fabes, Gradient estimates at the boundary for solutions to nondivergence elliptic equations, Harmonic analysis and partial differential equations (Boca Raton, FL, 1988), Contemp. Math., vol. 107, Amer. Math. Soc., Providence, RI, 1990, pp. 1–12. MR 1066466, 10.1090/conm/107/1066466
  • [4] E. B. Fabes and D. W. Stroock, The 𝐿^{𝑝}-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), no. 4, 997–1016. MR 771392, 10.1215/S0012-7094-84-05145-7
  • [5] Kaising Tso, On an Aleksandrov-Bakel′man type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations 10 (1985), no. 5, 543–553. MR 790223, 10.1080/03605308508820388
  • [6] N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Mat. Ž. 17 (1976), no. 2, 290–303, 478 (Russian). MR 0420016
  • [7] Carlo Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15–30 (Italian, with English summary). MR 0214905
  • [8] N. N. Ural'zeva and O. A. Ladyzhenskaya, A survey of results on the solvability of boundary value problems for second order uniformly elliptic and parabolic quasilinear equations having unbounded singularities, Russian Math. Surveys 41 (1986).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35K20, 35B05

Retrieve articles in all journals with MSC: 35K20, 35B05

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society