Title: Estimates Near the Boundary for Solutions of Second Order Parabolic Equations

We discuss different forms of the Harnack inequality for second order, linear, uniformly parabolic differential equations, and their applications to the estimates of solutions near the boundary. These applications include some Gaussian estimates and doubling properties for the caloric measure, and estimates for the quotient of two positive solutions vanishing on a portion of the boundary of a Lipschitz cylinder. A genera approach to all these problems is demonstrated, which works for both the divergence and non-divergence equations and is based only on the ``standard'' Harnack inequality and elementary comparison arguments.

1991 Mathematics Subject Classification: Primary 35K

Keywords and Phrases: Harnack inequality, Caloric measure, Doubling property

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