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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 $ 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 $.

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