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

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Abstract: We show that if is a solution to on a cylinder , where is a bounded open set in , and vanishes continuously on the parabolic boundary of . Then the maximum of on the cylinder is bounded by a constant depending on the ellipticity of the coefficient matrix , the diameter of , and the dimension times the norm of in .

**[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).

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1092918-X

Article copyright:
© Copyright 1992
American Mathematical Society