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

N. V. Krylov ^1,

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 and 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, Vol. 24 (2012), No. 1, 54-95.
[2]	N. V. Krylov, On Itô's stochastic integral equations, (Russian), Teoriya Veroyatnostei i eye Primeneniya, 14 (1969), 340-348; English translation in Theor. Probability Appl., 14 (1969), 330-336.
[3]	N. V. Krylov, Certain estimates in the theory of stochastic integrals, (Russian), Teoriya Veroyatnostei i eye Primeneniya, 18 (1973), 56-65; English translation in Theor. Probability Appl., 18 (1973), 54-63.
[4]	N. V. Krylov, Some estimates for the density of the distrinbution of a stochastic integral, (Russian), Izvestiya Akademii Nauk SSSR, seriya matematicheskaya, 38 (1974), 228-248; English translation in Math. USSR Izvestija, 8 (1974), 233-254.
[5]	N. V. Krylov, "Controlled Diffusion Processes,'' (Russian), Nauka, Moscow, 1977; English translation, Applications of Mathematics, 14, Springer-Verlag, New York-Berlin, 1980.
[6]	N. V. Krylov, "Nelineĭnye Éllipticheskie i Parabolicheskie Uravneniya Vtorogo Poryadka,'' (Russian) [Second-Order Nonlinear Elliptic and Parabolic Equations], "Nauka," Moscow, 1985; English translation, Reidel, Dordrecht, 1987, MR0901759.
[7]	N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces," Graduate Studies in Mathematics, 96, Amer. Math. Soc., Providence, RI, 2008.
[8]	N. V. Krylov, On Bellman's equations with VMO coefficients, Methods and Applications of Analysis, 17 (2010), 105-121.
[9]	Fang-Hua Lin, Second derivative $L^p$-estimates for elliptic equations of nondivergent type, Proc. Amer. Math. Soc., 96 (1986), 447-451. 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], "Nauka," Moscow, 1968; English translation, Amer. Math. Soc., Providence, RI, 1968.
[11]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 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, Vol. 24 (2012), No. 1, 54-95.
[2]	N. V. Krylov, On Itô's stochastic integral equations, (Russian), Teoriya Veroyatnostei i eye Primeneniya, 14 (1969), 340-348; English translation in Theor. Probability Appl., 14 (1969), 330-336.
[3]	N. V. Krylov, Certain estimates in the theory of stochastic integrals, (Russian), Teoriya Veroyatnostei i eye Primeneniya, 18 (1973), 56-65; English translation in Theor. Probability Appl., 18 (1973), 54-63.
[4]	N. V. Krylov, Some estimates for the density of the distrinbution of a stochastic integral, (Russian), Izvestiya Akademii Nauk SSSR, seriya matematicheskaya, 38 (1974), 228-248; English translation in Math. USSR Izvestija, 8 (1974), 233-254.
[5]	N. V. Krylov, "Controlled Diffusion Processes,'' (Russian), Nauka, Moscow, 1977; English translation, Applications of Mathematics, 14, Springer-Verlag, New York-Berlin, 1980.
[6]	N. V. Krylov, "Nelineĭnye Éllipticheskie i Parabolicheskie Uravneniya Vtorogo Poryadka,'' (Russian) [Second-Order Nonlinear Elliptic and Parabolic Equations], "Nauka," Moscow, 1985; English translation, Reidel, Dordrecht, 1987, MR0901759.
[7]	N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces," Graduate Studies in Mathematics, 96, Amer. Math. Soc., Providence, RI, 2008.
[8]	N. V. Krylov, On Bellman's equations with VMO coefficients, Methods and Applications of Analysis, 17 (2010), 105-121.
[9]	Fang-Hua Lin, Second derivative $L^p$-estimates for elliptic equations of nondivergent type, Proc. Amer. Math. Soc., 96 (1986), 447-451. 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], "Nauka," Moscow, 1968; English translation, Amer. Math. Soc., Providence, RI, 1968.
[11]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

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