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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

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, Vol. 24 (2012), No. 1, 54-95. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

N. V. Krylov, "Controlled Diffusion Processes,'' (Russian), Nauka, Moscow, 1977; English translation, Applications of Mathematics, 14, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[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.  Google Scholar

[7]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces," Graduate Studies in Mathematics, 96, Amer. Math. Soc., Providence, RI, 2008.  Google Scholar

[8]

N. V. Krylov, On Bellman's equations with VMO coefficients, Methods and Applications of Analysis, 17 (2010), 105-121.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

show all references

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

N. V. Krylov, "Controlled Diffusion Processes,'' (Russian), Nauka, Moscow, 1977; English translation, Applications of Mathematics, 14, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[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.  Google Scholar

[7]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces," Graduate Studies in Mathematics, 96, Amer. Math. Soc., Providence, RI, 2008.  Google Scholar

[8]

N. V. Krylov, On Bellman's equations with VMO coefficients, Methods and Applications of Analysis, 17 (2010), 105-121.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

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