Optimal regularity for parabolic Schrödinger operators

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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

Abstract
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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 and 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.
[2]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.
[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.
[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.
[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.
[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.
[7]	V. Kokilashvili and M. Krbec, "Weighted Inequalities in Lorentz and Orlicz Spaces," World Scientific, 1991. doi: 10.1142/1367.
[8]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[9]	W. Orlicz, Üeber eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220.
[10]	M. Rao and Z. Ren, "Applications of Orlicz Spaces," Marcel Dekker Inc., New York, 2000. doi: 10.1201/9780203910863.
[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.
[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.
[13]	E. M. Stein, "Harmonic Analysis," Princeton University Press, Princeton, 1993.
[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.
[15]	F. Yao, Optimal regularity for Schrödinger equations, Nonlinear Analysis, 71 (2009), 5144-5150. doi: 10.1016/j.na.2009.03.081.

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.
[2]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.
[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.
[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.
[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.
[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.
[7]	V. Kokilashvili and M. Krbec, "Weighted Inequalities in Lorentz and Orlicz Spaces," World Scientific, 1991. doi: 10.1142/1367.
[8]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[9]	W. Orlicz, Üeber eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220.
[10]	M. Rao and Z. Ren, "Applications of Orlicz Spaces," Marcel Dekker Inc., New York, 2000. doi: 10.1201/9780203910863.
[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.
[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.
[13]	E. M. Stein, "Harmonic Analysis," Princeton University Press, Princeton, 1993.
[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.
[15]	F. Yao, Optimal regularity for Schrödinger equations, Nonlinear Analysis, 71 (2009), 5144-5150. doi: 10.1016/j.na.2009.03.081.

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