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May  2013, 12(3): 1407-1414. doi: 10.3934/cpaa.2013.12.1407

Optimal regularity for parabolic Schrödinger operators

Fengping Yao 1,

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  July 2011 Revised  August 2012 Published  September 2012

In this paper we study the regularity theory for the parabolic Schrödinger operator $P=\frac{\partial}{\partial t}-\triangle+V$ under optimal conditions. As a corollary we obtain $L^p$-type regularity estimates for such operator.
Keywords: Schrödinger operator, regularity, optimal, parabolic, Orlicz spaces..
Mathematics Subject Classification: 35J10; 35K1.
Citation: Fengping Yao. Optimal regularity for parabolic Schrödinger operators. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934/cpaa.2013.12.1407
References:
[1]

E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320. doi: 10.1215/S0012-7094-07-13623-8.  Google Scholar

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R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.  Google Scholar

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A. Benkirane and A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces,, \emph{ \textcolor{blue}{in Orlicz spaces}, ().  doi: 10.1016/S0362-546X(97)00612-3.  Google Scholar

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S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations, J. Funct. Anal., 255 (2008), 1851-1873. doi: 10.1016/j.jfa.2008.09.007.  Google Scholar

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A. Carbonaro, G. Metafune and C. Spina, Parabolic Schrödinger operators, J. Math. Anal. Appl., 343 (2008), 965-974. doi: 10.1016/j.jmaa.2008.02.010.  Google Scholar

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W. Gao and Y. Jiang, $L^p$ estimate for parabolic Schrödinger operator with certain potentials, J. Math. Anal. Appl., 310 (2005), 128-143. doi: 10.1016/j.jmaa.2005.01.049.  Google Scholar

[7]

V. Kokilashvili and M. Krbec, "Weighted Inequalities in Lorentz and Orlicz Spaces," World Scientific, 1991. doi: 10.1142/1367.  Google Scholar

[8]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[9]

W. Orlicz, Üeber eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220. Google Scholar

[10]

M. Rao and Z. Ren, "Applications of Orlicz Spaces," Marcel Dekker Inc., New York, 2000. doi: 10.1201/9780203910863.  Google Scholar

[11]

Z. Shen, On the Neumann problem for Schrödinger operators in Lipschitz domains, Indiana Univ. Math. J., 43 (1994), 143-176. doi: 10.1512/iumj.1994.43.43007.  Google Scholar

[12]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546. doi: 10.5802/aif.1463.  Google Scholar

[13]

E. M. Stein, "Harmonic Analysis," Princeton University Press, Princeton, 1993.  Google Scholar

[14]

L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation, Proc. Amer. Math. Soc., 137 (2009), 2037-2047. doi: 10.1090/S0002-9939-09-09805-0.  Google Scholar

[15]

F. Yao, Optimal regularity for Schrödinger equations, Nonlinear Analysis, 71 (2009), 5144-5150. doi: 10.1016/j.na.2009.03.081.  Google Scholar

show all references

References:
[1]

E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320. doi: 10.1215/S0012-7094-07-13623-8.  Google Scholar

[2]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.  Google Scholar

[3]

A. Benkirane and A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces,, \emph{ \textcolor{blue}{in Orlicz spaces}, ().  doi: 10.1016/S0362-546X(97)00612-3.  Google Scholar

[4]

S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations, J. Funct. Anal., 255 (2008), 1851-1873. doi: 10.1016/j.jfa.2008.09.007.  Google Scholar

[5]

A. Carbonaro, G. Metafune and C. Spina, Parabolic Schrödinger operators, J. Math. Anal. Appl., 343 (2008), 965-974. doi: 10.1016/j.jmaa.2008.02.010.  Google Scholar

[6]

W. Gao and Y. Jiang, $L^p$ estimate for parabolic Schrödinger operator with certain potentials, J. Math. Anal. Appl., 310 (2005), 128-143. doi: 10.1016/j.jmaa.2005.01.049.  Google Scholar

[7]

V. Kokilashvili and M. Krbec, "Weighted Inequalities in Lorentz and Orlicz Spaces," World Scientific, 1991. doi: 10.1142/1367.  Google Scholar

[8]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[9]

W. Orlicz, Üeber eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220. Google Scholar

[10]

M. Rao and Z. Ren, "Applications of Orlicz Spaces," Marcel Dekker Inc., New York, 2000. doi: 10.1201/9780203910863.  Google Scholar

[11]

Z. Shen, On the Neumann problem for Schrödinger operators in Lipschitz domains, Indiana Univ. Math. J., 43 (1994), 143-176. doi: 10.1512/iumj.1994.43.43007.  Google Scholar

[12]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546. doi: 10.5802/aif.1463.  Google Scholar

[13]

E. M. Stein, "Harmonic Analysis," Princeton University Press, Princeton, 1993.  Google Scholar

[14]

L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation, Proc. Amer. Math. Soc., 137 (2009), 2037-2047. doi: 10.1090/S0002-9939-09-09805-0.  Google Scholar

[15]

F. Yao, Optimal regularity for Schrödinger equations, Nonlinear Analysis, 71 (2009), 5144-5150. doi: 10.1016/j.na.2009.03.081.  Google Scholar

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