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Two Results Relating an $L^p$ Regularity Condition and the $L^q$ Dirichlet Problem for Parabolic Equations

Commun. Math. Anal.
Volume 22, Number 2 (2019), 35 - 60

Two Results Relating an $L^p$ Regularity Condition and the $L^q$ Dirichlet Problem for Parabolic Equations

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Abstract

We consider variations and generalizations of the initial Dirichlet problem for linear second order divergence form equations of parabolic type, with vanishing initial values and non-continuous lateral data, in the setting of Lipschitz cylinders. More precisely, lateral data in adequations of the Lebesgue classes $L^p$, and a family of Sobolev-type classes are considered. We also establish some basic connections between estimates related to solvability of each of these problems. This generalizes some of the well-known works for Laplace's equation, heat equation and some linear elliptic-type equations of second order.