Elliptic and Parabolic Equations - Abstract

Kozlov, Vladimir

The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients

In 2001, N. Krylov observed that for non-divergence parabolic equations coercive estimates for solutions can be proved even when the leading coefficients are only measurable functions with respect to $t$. In this lecture I give an overview of results obtained in this direction and present new ones obtained together with Alexander Nazarov. We consider the Dirichlet problem for non-divergence parabolic equation with discontinuous in $t$ coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi-linear parabolic equations in a bounded domain. In particular, if the boundary is of class $C^1,delta, deltain[0, 1]$, then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary.