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November  2014, 13(6): 2253-2272. doi: 10.3934/cpaa.2014.13.2253

Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case

P. Cerejeiras 1, , U. Kähler 2, , M. M. Rodrigues 2, and  N. Vieira 2,

1. 

Department of Mathematics, University of Aveiro, P-3810-193 Aveiro
2. 

CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal, Portugal, Portugal

Received  June 2013 Revised  February 2014 Published  July 2014

In this work, we present a Hodge-type decomposition for variable exponent spaces of Clifford-valued functions, where one of the components is the kernel of the parabolic-type Dirac operator.
Keywords: Schrödinger equation, Witt basis, parabolic-type Dirac operator, fundamental solution, regularization procedure, variable exponent spaces, Hodge decomposition., time dependent operators, Clifford analysis.
Mathematics Subject Classification: Primary: 30G35; Secondary: 35Q41, 35A08, 46E35, 46E30, 34L4.
Citation: P. Cerejeiras, U. Kähler, M. M. Rodrigues, N. Vieira. Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253
References:
[1]

A. Almeida and P. Hästö, Interpolation in variable exponent spaces,, \emph{Rev. Mat. Complut.}, ().  doi: 10.1007/s13163-013-0135-1.  Google Scholar

[2]

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids,, \emph{Arch. Ration. Mech. Anal.}, 164 (2002), 213.  doi: 10.1007/s00205-002-0208-7.  Google Scholar

[3]

R. Artino and J. Barros-Neto, Hypoelliptic Boundary-value Problems,, Lectures Notes in Pure and Applied Mathematics-Vol.53, (1980).   Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation spaces. An introduction,, Grundlehren der mathematischen Wissenschaften-Vol.223, (1976).   Google Scholar

[5]

P. Cerejeiras and N. Vieira, Regularization of the non-stationary Schrödinger operator,, \emph{Math. Meth. in Appl. Sc.}, 32 (2009), 535.  doi: 10.1002/mma.1050.  Google Scholar

[6]

P. Cerejeiras and N. Vieira, Factorization of the non-stationary Schrödinger operator,, \emph{Adv. Appl. Clifford Algebr.}, 17 (2007), 331.  doi: 10.1007/s00006-007-0039-6.  Google Scholar

[7]

P. Cerejeiras, U. Kähler and F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains,, \emph{Math. Meth. in Appl. Sc.}, 28 (2005), 1715.  doi: 10.1002/mma.634.  Google Scholar

[8]

Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383.  doi: 10.1137/050624522.  Google Scholar

[9]

R. Delanghe, F. Sommen and V. Souček, Clifford Algebras and Spinor-valued Functions. A Function Theory for the Dirac Operator,, Mathematics and its Applications-Vol.53, (1992).  doi: 10.1007/978-94-011-2922-0.  Google Scholar

[10]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolec Spaces with Variable Exponents,, Springer-Verlang, (2011).  doi: 10.1007/978-3-642-18363-8.  Google Scholar

[11]

L. Diening, D. Lengeler and M. Ružička, The Stokes and Poisson problem in variable exponents spaces,, \emph{Complex Var. Elliptic Equ.}, 56 (2011), 789.  doi: 10.1080/17476933.2010.504843.  Google Scholar

[12]

R. Fortini, D. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities,, \emph{Nonlinear Anal.}, 70 (2009), 2917.  doi: 10.1016/j.na.2008.12.030.  Google Scholar

[13]

K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers,, Mathematical Methods in Practice, (1997).   Google Scholar

[14]

L. Hormander, On the regularity of the solutions of boundary problems,, \emph{Acta. Math.}, 99 (1958), 225.   Google Scholar

[15]

R. F. Hoskins and J.S. Pinto, Theories of Generalised Functions - Distributions, Ultradistributions and other Generalised Functions,, Horwood Publishing, (2005).  doi: 10.1533/9780857099488.  Google Scholar

[16]

T. Kato, Nonlinear Schrödinger equation,, in \emph{Schr\, 345 (1989).  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[17]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592.   Google Scholar

[18]

R. S. Kraußhar and N. Vieira, The Schrödinger equation on cylinders and the $n$-torus,, \emph{J. Evol. Equ.}, 11 (2011), 215.  doi: 10.1007/s00028-010-0089-4.  Google Scholar

[19]

N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002).  doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[20]

N. Laskin, Fractional quantum mechanics,, \emph{Phy. Rev. E}, 62 (2000), 3135.   Google Scholar

[21]

N. Laskin, Fractional quantum mechanics and Lévy path integrals,, \emph{Phys. Lett. A}, 268 (2000), 298.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[22]

S. G. Mikhlin and S. Prössdorf, Singular Integral Operators,, Springer-Verlag, (1986).  doi: 10.1007/978-3-642-61631-0.  Google Scholar

[23]

H. Nakano, Modulared Semi-Ordered Linear Spaces,, Maruzen Co. Ltd., (1950).   Google Scholar

[24]

H. Nakano, Topology of Linear Topological Spaces,, Maruzen Co. Ltd., (1951).   Google Scholar

[25]

M. Sanchón and J. M. Urbano, Entropy solutions for the p(x)-Laplace equation,, \emph{Trans. Amer. Math. Soc.}, 361 (2009), 6387.  doi: 10.1090/S0002-9947-09-04399-2.  Google Scholar

[26]

S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,, \emph{Integr. Transf. Spec. F.}, 16 (2005), 461.  doi: 10.1080/10652460412331320322.  Google Scholar

[27]

W. Sprößig, On Helmotz decompositions and their generalizations-an overview,, \emph{Math. Meth. in Appl. Sc.}, 33 (2009), 374.  doi: 10.1002/mma.1212.  Google Scholar

[28]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics-Vol.106, (2006).   Google Scholar

[29]

G. Velo, Mathematical Aspects of the nonlinear Schrödinger Equation,, in \emph{Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrdinger systems: theory and applications} (eds. L. V\'azquez et al.), (1996), 39.   Google Scholar

show all references

References:
[1]

A. Almeida and P. Hästö, Interpolation in variable exponent spaces,, \emph{Rev. Mat. Complut.}, ().  doi: 10.1007/s13163-013-0135-1.  Google Scholar

[2]

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids,, \emph{Arch. Ration. Mech. Anal.}, 164 (2002), 213.  doi: 10.1007/s00205-002-0208-7.  Google Scholar

[3]

R. Artino and J. Barros-Neto, Hypoelliptic Boundary-value Problems,, Lectures Notes in Pure and Applied Mathematics-Vol.53, (1980).   Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation spaces. An introduction,, Grundlehren der mathematischen Wissenschaften-Vol.223, (1976).   Google Scholar

[5]

P. Cerejeiras and N. Vieira, Regularization of the non-stationary Schrödinger operator,, \emph{Math. Meth. in Appl. Sc.}, 32 (2009), 535.  doi: 10.1002/mma.1050.  Google Scholar

[6]

P. Cerejeiras and N. Vieira, Factorization of the non-stationary Schrödinger operator,, \emph{Adv. Appl. Clifford Algebr.}, 17 (2007), 331.  doi: 10.1007/s00006-007-0039-6.  Google Scholar

[7]

P. Cerejeiras, U. Kähler and F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains,, \emph{Math. Meth. in Appl. Sc.}, 28 (2005), 1715.  doi: 10.1002/mma.634.  Google Scholar

[8]

Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383.  doi: 10.1137/050624522.  Google Scholar

[9]

R. Delanghe, F. Sommen and V. Souček, Clifford Algebras and Spinor-valued Functions. A Function Theory for the Dirac Operator,, Mathematics and its Applications-Vol.53, (1992).  doi: 10.1007/978-94-011-2922-0.  Google Scholar

[10]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolec Spaces with Variable Exponents,, Springer-Verlang, (2011).  doi: 10.1007/978-3-642-18363-8.  Google Scholar

[11]

L. Diening, D. Lengeler and M. Ružička, The Stokes and Poisson problem in variable exponents spaces,, \emph{Complex Var. Elliptic Equ.}, 56 (2011), 789.  doi: 10.1080/17476933.2010.504843.  Google Scholar

[12]

R. Fortini, D. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities,, \emph{Nonlinear Anal.}, 70 (2009), 2917.  doi: 10.1016/j.na.2008.12.030.  Google Scholar

[13]

K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers,, Mathematical Methods in Practice, (1997).   Google Scholar

[14]

L. Hormander, On the regularity of the solutions of boundary problems,, \emph{Acta. Math.}, 99 (1958), 225.   Google Scholar

[15]

R. F. Hoskins and J.S. Pinto, Theories of Generalised Functions - Distributions, Ultradistributions and other Generalised Functions,, Horwood Publishing, (2005).  doi: 10.1533/9780857099488.  Google Scholar

[16]

T. Kato, Nonlinear Schrödinger equation,, in \emph{Schr\, 345 (1989).  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[17]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$,, \emph{Czechoslovak Math. J.}, 41 (1991), 592.   Google Scholar

[18]

R. S. Kraußhar and N. Vieira, The Schrödinger equation on cylinders and the $n$-torus,, \emph{J. Evol. Equ.}, 11 (2011), 215.  doi: 10.1007/s00028-010-0089-4.  Google Scholar

[19]

N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002).  doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[20]

N. Laskin, Fractional quantum mechanics,, \emph{Phy. Rev. E}, 62 (2000), 3135.   Google Scholar

[21]

N. Laskin, Fractional quantum mechanics and Lévy path integrals,, \emph{Phys. Lett. A}, 268 (2000), 298.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[22]

S. G. Mikhlin and S. Prössdorf, Singular Integral Operators,, Springer-Verlag, (1986).  doi: 10.1007/978-3-642-61631-0.  Google Scholar

[23]

H. Nakano, Modulared Semi-Ordered Linear Spaces,, Maruzen Co. Ltd., (1950).   Google Scholar

[24]

H. Nakano, Topology of Linear Topological Spaces,, Maruzen Co. Ltd., (1951).   Google Scholar

[25]

M. Sanchón and J. M. Urbano, Entropy solutions for the p(x)-Laplace equation,, \emph{Trans. Amer. Math. Soc.}, 361 (2009), 6387.  doi: 10.1090/S0002-9947-09-04399-2.  Google Scholar

[26]

S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,, \emph{Integr. Transf. Spec. F.}, 16 (2005), 461.  doi: 10.1080/10652460412331320322.  Google Scholar

[27]

W. Sprößig, On Helmotz decompositions and their generalizations-an overview,, \emph{Math. Meth. in Appl. Sc.}, 33 (2009), 374.  doi: 10.1002/mma.1212.  Google Scholar

[28]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics-Vol.106, (2006).   Google Scholar

[29]

G. Velo, Mathematical Aspects of the nonlinear Schrödinger Equation,, in \emph{Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrdinger systems: theory and applications} (eds. L. V\'azquez et al.), (1996), 39.   Google Scholar

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