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May  2016, 15(3): 761-794. doi: 10.3934/cpaa.2016.15.761

A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

1. 

Department of mathematics, Korea university, 1 anam-dong, sungbuk-gu, Seoul 136-701, South Korea

2. 

Department of Mathematics, Ajou University, 206 Worldcup-ro, Yeontong-gu, Suwon 443-749, South Korea

Received  March 2015 Revised  December 2015 Published  February 2015

In this article we consider parabolic systems and $L_p$ regularity of the solutions. With zero boundary condition the solutions experience bad regularity near the boundary. This article addresses a possible way of describing the regularity nature. Our space domain is a half space and we adapt an appropriate weight into our function spaces. In this weighted Sobolev space setting we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations. Using these, we prove uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
Citation: Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
References:
[1]

M. Bramanti and M. C. Ceruti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients,, \emph{Comm. Partial Differential Equations}, 18 (1993), 1735. doi: 10.1080/03605309308820991. Google Scholar

[2]

Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains,, \emph{J. Differential Equations}, 209 (2005), 229. doi: 10.1016/j.jde.2004.08.018. Google Scholar

[3]

F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,, \emph{Trans. Amer. Math. Soc.}, 336 (1993), 841. doi: 10.2307/2154379. Google Scholar

[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics \textbf{24}, 24 (1985). Google Scholar

[5]

R. Haller-Dintelmann, H. Heck and M. Hieber, $L^p-L^q$-estimates for parabolic systems in non-divergence form with VMO coefficients,, \emph{J. London Math. Soc.}, 74 (2006), 717. doi: 10.1112/S0024610706023192. Google Scholar

[6]

Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients,, \emph{Potential Anal.}, 26 (2007), 345. doi: 10.1007/s11118-007-9042-8. Google Scholar

[7]

N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces,, \emph{Journal of Functional Analysis}, 183 (2001), 1. doi: 10.1006/jfan.2000.3728. Google Scholar

[8]

N. V. Krylov, The heat equation in $L_p((0,T),L_p)$-spaces with weights,, \emph{SIAM J. Math. Anal.}, 32 (2001), 1117. doi: 10.1137/S0036141000372039. Google Scholar

[9]

N. V. Krylov, An analytic approach to SPDEs,, in \emph{Stochastic Partial Differential Equations: Six Perspectives}, 64 (1999), 185. doi: 10.1090/surv/064/05. Google Scholar

[10]

N. V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space,, \emph{Comm. in PDEs}, 23 (1999), 1611. doi: 10.1080/03605309908821478. Google Scholar

[11]

N. V. Krylov, Some properties of weighted Sobolev spaces in $\bR^d_+$,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 28 (1999), 675. Google Scholar

[12]

N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains,, \emph{Probab. Theory Relat. Fields}, 98 (1994), 389. doi: 10.1007/BF01192260. Google Scholar

[13]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,, American Mathematical Society, (2008). doi: 10.1090/gsm/096. Google Scholar

[14]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line,, \emph{SIAM J. Math. Anal.}, 30 (1999), 298. doi: 10.1137/S0036141097326908. Google Scholar

[15]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space,, \emph{SIAM J. on Math. Anal.}, 31 (1999), 19. doi: 10.1137/S0036141098338843. Google Scholar

[16]

Kijung Lee, On a deterministic linear partial differential system,, \emph{J. Math. Anal. Appl.}, 353 (2009), 24. doi: 10.1016/j.jmaa.2008.11.059. Google Scholar

[17]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$,, Dunod, (1968). Google Scholar

[18]

S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations,, \emph{Methods and Applications of Analysis}, 1 (2000), 195. Google Scholar

[19]

V. A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle,, \emph{Zapiski Nauchnykh Seminarov LOMI}, 138 (1984), 146. Google Scholar

[20]

H. Triebel, Theory of Function Spaces,, Birkh\, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar

show all references

References:
[1]

M. Bramanti and M. C. Ceruti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients,, \emph{Comm. Partial Differential Equations}, 18 (1993), 1735. doi: 10.1080/03605309308820991. Google Scholar

[2]

Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains,, \emph{J. Differential Equations}, 209 (2005), 229. doi: 10.1016/j.jde.2004.08.018. Google Scholar

[3]

F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,, \emph{Trans. Amer. Math. Soc.}, 336 (1993), 841. doi: 10.2307/2154379. Google Scholar

[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics \textbf{24}, 24 (1985). Google Scholar

[5]

R. Haller-Dintelmann, H. Heck and M. Hieber, $L^p-L^q$-estimates for parabolic systems in non-divergence form with VMO coefficients,, \emph{J. London Math. Soc.}, 74 (2006), 717. doi: 10.1112/S0024610706023192. Google Scholar

[6]

Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients,, \emph{Potential Anal.}, 26 (2007), 345. doi: 10.1007/s11118-007-9042-8. Google Scholar

[7]

N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces,, \emph{Journal of Functional Analysis}, 183 (2001), 1. doi: 10.1006/jfan.2000.3728. Google Scholar

[8]

N. V. Krylov, The heat equation in $L_p((0,T),L_p)$-spaces with weights,, \emph{SIAM J. Math. Anal.}, 32 (2001), 1117. doi: 10.1137/S0036141000372039. Google Scholar

[9]

N. V. Krylov, An analytic approach to SPDEs,, in \emph{Stochastic Partial Differential Equations: Six Perspectives}, 64 (1999), 185. doi: 10.1090/surv/064/05. Google Scholar

[10]

N. V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space,, \emph{Comm. in PDEs}, 23 (1999), 1611. doi: 10.1080/03605309908821478. Google Scholar

[11]

N. V. Krylov, Some properties of weighted Sobolev spaces in $\bR^d_+$,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 28 (1999), 675. Google Scholar

[12]

N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains,, \emph{Probab. Theory Relat. Fields}, 98 (1994), 389. doi: 10.1007/BF01192260. Google Scholar

[13]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,, American Mathematical Society, (2008). doi: 10.1090/gsm/096. Google Scholar

[14]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line,, \emph{SIAM J. Math. Anal.}, 30 (1999), 298. doi: 10.1137/S0036141097326908. Google Scholar

[15]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space,, \emph{SIAM J. on Math. Anal.}, 31 (1999), 19. doi: 10.1137/S0036141098338843. Google Scholar

[16]

Kijung Lee, On a deterministic linear partial differential system,, \emph{J. Math. Anal. Appl.}, 353 (2009), 24. doi: 10.1016/j.jmaa.2008.11.059. Google Scholar

[17]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$,, Dunod, (1968). Google Scholar

[18]

S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations,, \emph{Methods and Applications of Analysis}, 1 (2000), 195. Google Scholar

[19]

V. A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle,, \emph{Zapiski Nauchnykh Seminarov LOMI}, 138 (1984), 146. Google Scholar

[20]

H. Triebel, Theory of Function Spaces,, Birkh\, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar

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