American Institute of Mathematical Sciences

Advanced
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, (2012), 54.   Google Scholar

[2]

N. V. Krylov, On Itô's stochastic integral equations,, (Russian), 14 (1969), 340.   Google Scholar

[3]

N. V. Krylov, Certain estimates in the theory of stochastic integrals,, (Russian), 18 (1973), 56.   Google Scholar

[4]

N. V. Krylov, Some estimates for the density of the distrinbution of a stochastic integral,, (Russian), 38 (1974), 228.   Google Scholar

[5]

N. V. Krylov, "Controlled Diffusion Processes,'', (Russian), 14 (1977).   Google Scholar

[6]

N. V. Krylov, "Nelineĭnye Éllipticheskie i Parabolicheskie Uravneniya Vtorogo Poryadka,'', (Russian) [Second-Order Nonlinear Elliptic and Parabolic Equations], (1985).   Google Scholar

[7]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,", Graduate Studies in Mathematics, 96 (2008).   Google Scholar

[8]

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

[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.  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], (1968).   Google Scholar

[11]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (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, (2012), 54.   Google Scholar

[2]

N. V. Krylov, On Itô's stochastic integral equations,, (Russian), 14 (1969), 340.   Google Scholar

[3]

N. V. Krylov, Certain estimates in the theory of stochastic integrals,, (Russian), 18 (1973), 56.   Google Scholar

[4]

N. V. Krylov, Some estimates for the density of the distrinbution of a stochastic integral,, (Russian), 38 (1974), 228.   Google Scholar

[5]

N. V. Krylov, "Controlled Diffusion Processes,'', (Russian), 14 (1977).   Google Scholar

[6]

N. V. Krylov, "Nelineĭnye Éllipticheskie i Parabolicheskie Uravneniya Vtorogo Poryadka,'', (Russian) [Second-Order Nonlinear Elliptic and Parabolic Equations], (1985).   Google Scholar

[7]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,", Graduate Studies in Mathematics, 96 (2008).   Google Scholar

[8]

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

[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.  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], (1968).   Google Scholar

[11]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).   Google Scholar

[1]

Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121
[2]

Ugur G. Abdulla. Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3379-3397. doi: 10.3934/dcds.2012.32.3379
[3]

Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations & Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61
[4]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[5]

Luigi Greco, Gioconda Moscariello, Teresa Radice. Nondivergence elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 131-143. doi: 10.3934/dcdsb.2009.11.131
[6]

Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241
[7]

Giorgio Metafune, Chiara Spina, Cristian Tacelli. On a class of elliptic operators with unbounded diffusion coefficients. Evolution Equations & Control Theory, 2014, 3 (4) : 671-680. doi: 10.3934/eect.2014.3.671
[8]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155
[9]

Kenneth Kuttler. Measurable solutions for elliptic and evolution inclusions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020041
[10]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[11]

N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119
[12]

Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285
[13]

Fuzhi Li, Yangrong Li, Renhai Wang. Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3663-3685. doi: 10.3934/dcds.2018158
[14]

Dong Sun, V. S. Manoranjan, Hong-Ming Yin. Numerical solutions for a coupled parabolic equations arising induction heating processes. Conference Publications, 2007, 2007 (Special) : 956-964. doi: 10.3934/proc.2007.2007.956
[15]

Pierre-A. Vuillermot. On the time evolution of Bernstein processes associated with a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1073-1090. doi: 10.3934/dcdsb.2018142
[16]

Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17
[17]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130
[18]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[19]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[20]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (114)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences