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

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September 2012, 17(6): 2073-2090. doi: 10.3934/dcdsb.2012.17.2073

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

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

Received May 2011 Revised January 2012 Published May 2012

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

Mathematics Subject Classification: 35J15, 35K10, 60H1.

Citation: N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073

References:

[1]	Hongjie Dong, N. V. Krylov and Xu Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients,, Algebra i Analiz, (2012), 54.
[2]	N. V. Krylov, On Itô's stochastic integral equations,, (Russian), 14 (1969), 340.
[3]	N. V. Krylov, Certain estimates in the theory of stochastic integrals,, (Russian), 18 (1973), 56.
[4]	N. V. Krylov, Some estimates for the density of the distrinbution of a stochastic integral,, (Russian), 38 (1974), 228.
[5]	N. V. Krylov, "Controlled Diffusion Processes,'', (Russian), 14 (1977).
[6]	N. V. Krylov, "Nelineĭnye Éllipticheskie i Parabolicheskie Uravneniya Vtorogo Poryadka,'', (Russian) [Second-Order Nonlinear Elliptic and Parabolic Equations], (1985).
[7]	N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,", Graduate Studies in Mathematics, 96 (2008).
[8]	N. V. Krylov, On Bellman's equations with VMO coefficients,, Methods and Applications of Analysis, 17 (2010), 105.
[9]	Fang-Hua Lin, Second derivative $L^p$-estimates for elliptic equations of nondivergent type,, Proc. Amer. Math. Soc., 96 (1986), 447. doi: 10.2307/2046592.
[10]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, "Lineĭnye i Kvazilineĭnye Uravneniya Parabolicheskogo Tipa,'', (Russian) [Linear and Quasi-Linear Equations of Parabolic Type], (1968).
[11]	G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).

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References:

[1]	Hongjie Dong, N. V. Krylov and Xu Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients,, Algebra i Analiz, (2012), 54.
[2]	N. V. Krylov, On Itô's stochastic integral equations,, (Russian), 14 (1969), 340.
[3]	N. V. Krylov, Certain estimates in the theory of stochastic integrals,, (Russian), 18 (1973), 56.
[4]	N. V. Krylov, Some estimates for the density of the distrinbution of a stochastic integral,, (Russian), 38 (1974), 228.
[5]	N. V. Krylov, "Controlled Diffusion Processes,'', (Russian), 14 (1977).
[6]	N. V. Krylov, "Nelineĭnye Éllipticheskie i Parabolicheskie Uravneniya Vtorogo Poryadka,'', (Russian) [Second-Order Nonlinear Elliptic and Parabolic Equations], (1985).
[7]	N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,", Graduate Studies in Mathematics, 96 (2008).
[8]	N. V. Krylov, On Bellman's equations with VMO coefficients,, Methods and Applications of Analysis, 17 (2010), 105.
[9]	Fang-Hua Lin, Second derivative $L^p$-estimates for elliptic equations of nondivergent type,, Proc. Amer. Math. Soc., 96 (1986), 447. doi: 10.2307/2046592.
[10]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, "Lineĭnye i Kvazilineĭnye Uravneniya Parabolicheskogo Tipa,'', (Russian) [Linear and Quasi-Linear Equations of Parabolic Type], (1968).
[11]	G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).

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