Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients

N. V. Krylov

doi:10.3934/dcdsb.2012.17.2073

Article Contents

2012, Volume 17, Issue 6: 2073-2090. Doi: 10.3934/dcdsb.2012.17.2073

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Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients

N. V. Krylov^1,

1.
127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455

Received: April 30, 2011

Revised: December 31, 2011

Published: August 2012

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Abstract

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order equations in [1]. The first type is an estimate of the $\gamma$th norm of the second-order derivatives, where $\gamma\in(0,1)$, and the second type deals with estimates of the resolvent operators in $L_{p}$ when the first-order coefficients are summable to an appropriate power.

Keywords:

Elliptic and parabolic equations,
measurable coefficients,
occupation measures for diffusion processes.

Mathematics Subject Classification: 35J15, 35K10, 60H10.

Citation:

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References

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