Optimal regularity for parabolic Schrödinger operators

Previous Article
Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems
CPAA Home
This Issue
Next Article
Controllability results for a class of one dimensional degenerate/singular parabolic equations

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

Optimal regularity for parabolic Schrödinger operators

1.	Department of Mathematics, Shanghai University, Shanghai 200444, China

Received July 2011 Revised August 2012 Published September 2012

Abstract
Related Papers

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. 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, (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. 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. 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. 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., (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. Google Scholar
[10]	M. Rao and Z. Ren, "Applications of Orlicz Spaces,", Marcel Dekker Inc., (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. 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. doi: 10.5802/aif.1463. Google Scholar
[13]	E. M. Stein, "Harmonic Analysis,", Princeton University Press, (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. doi: 10.1090/S0002-9939-09-09805-0. Google Scholar
[15]	F. Yao, Optimal regularity for Schrödinger equations,, Nonlinear Analysis, 71 (2009), 5144. 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. 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, (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. 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. 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. 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., (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. Google Scholar
[10]	M. Rao and Z. Ren, "Applications of Orlicz Spaces,", Marcel Dekker Inc., (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. 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. doi: 10.5802/aif.1463. Google Scholar
[13]	E. M. Stein, "Harmonic Analysis,", Princeton University Press, (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. doi: 10.1090/S0002-9939-09-09805-0. Google Scholar
[15]	F. Yao, Optimal regularity for Schrödinger equations,, Nonlinear Analysis, 71 (2009), 5144. doi: 10.1016/j.na.2009.03.081. Google Scholar

[1]	Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435
[2]	Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407
[3]	Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213
[4]	Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
[5]	Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157
[6]	Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
[7]	Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125
[8]	Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061
[9]	Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[10]	Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[11]	Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933
[12]	Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
[13]	Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[14]	Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001
[15]	Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
[16]	Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems & Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055
[17]	Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[18]	Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[19]	Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[20]	Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences