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# A Note on Krylov-Tso's Parabolic Inequality

Luis Escauriaza
Proceedings of the American Mathematical Society
Vol. 115, No. 4 (Aug., 1992), pp. 1053-1056
DOI: 10.2307/2159354
Stable URL: http://www.jstor.org/stable/2159354
Page Count: 4
We show that if u is a solution to ∑n i, j = 1 aij(x, t)Diju(x, t) - Dtu(x, t) = φ(x) on a cylinder ΩT = Ω × (0, T), where Ω is a bounded open set in $\mathbf{R}^n, T > 0$, and u vanishes continuously on the parabolic boundary of ΩT. Then the maximum of u on the cylinder is bounded by a constant C depending on the ellipticity of the coefficient matrix (aij(x, t)), the diameter of Ω, and the dimension n times the Ln norm of φ in Ω.