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

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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

Abstract
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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 and 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, Rev. Mat. Complut., in press. doi: 10.1007/s13163-013-0135-1.
[2]	E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. doi: 10.1007/s00205-002-0208-7.
[3]	R. Artino and J. Barros-Neto, Hypoelliptic Boundary-value Problems, Lectures Notes in Pure and Applied Mathematics-Vol.53, Marcel Dekker, New York-Basel, 1980.
[4]	J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der mathematischen Wissenschaften-Vol.223, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[5]	P. Cerejeiras and N. Vieira, Regularization of the non-stationary Schrödinger operator, Math. Meth. in Appl. Sc., 32 (2009), 535-555. doi: 10.1002/mma.1050.
[6]	P. Cerejeiras and N. Vieira, Factorization of the non-stationary Schrödinger operator, Adv. Appl. Clifford Algebr., 17 (2007), 331-341. doi: 10.1007/s00006-007-0039-6.
[7]	P. Cerejeiras, U. Kähler and F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains, Math. Meth. in Appl. Sc., 28 (2005), 1715-1724. doi: 10.1002/mma.634.
[8]	Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.
[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, Kluwer Academic Publishers, Dordrecht etc., 1992. doi: 10.1007/978-94-011-2922-0.
[10]	L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolec Spaces with Variable Exponents, Springer-Verlang, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.
[11]	L. Diening, D. Lengeler and M. Ružička, The Stokes and Poisson problem in variable exponents spaces, Complex Var. Elliptic Equ., 56 (2011), 789-811. doi: 10.1080/17476933.2010.504843.
[12]	R. Fortini, D. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities, Nonlinear Anal., 70 (2009), 2917-2929. doi: 10.1016/j.na.2008.12.030.
[13]	K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Mathematical Methods in Practice, Wiley, Chichester, 1997.
[14]	L. Hormander, On the regularity of the solutions of boundary problems, Acta. Math., 99 (1958), 225-264.
[15]	R. F. Hoskins and J.S. Pinto, Theories of Generalised Functions - Distributions, Ultradistributions and other Generalised Functions, Horwood Publishing, Chichester, 2005. doi: 10.1533/9780857099488.
[16]	T. Kato, Nonlinear Schrödinger equation, in Schrödinger Operators, (eds. H. Holden and A. Jensen), Lectures Notes in Physics 345, Springer, Berlin, 1989. doi: 10.1007/3-540-51783-9_22.
[17]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
[18]	R. S. Kraußhar and N. Vieira, The Schrödinger equation on cylinders and the $n$-torus, J. Evol. Equ., 11 (2011), 215-237. doi: 10.1007/s00028-010-0089-4.
[19]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.
[20]	N. Laskin, Fractional quantum mechanics, Phy. Rev. E, 62 (2000), 3135-3145.
[21]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[22]	S. G. Mikhlin and S. Prössdorf, Singular Integral Operators, Springer-Verlag, Berlin etc., 1986. doi: 10.1007/978-3-642-61631-0.
[23]	H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950.
[24]	H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951.
[25]	M. Sanchón and J. M. Urbano, Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc., 361 (2009), 6387-6405. doi: 10.1090/S0002-9947-09-04399-2.
[26]	S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integr. Transf. Spec. F., 16 (2005), 461-482. doi: 10.1080/10652460412331320322.
[27]	W. Sprößig, On Helmotz decompositions and their generalizations-an overview, Math. Meth. in Appl. Sc., 33 (2009), 374-383. doi: 10.1002/mma.1212.
[28]	T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics-Vol.106, American Mathematical Society, 2006.
[29]	G. Velo, Mathematical Aspects of the nonlinear Schrödinger Equation, in Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrdinger systems: theory and applications (eds. L. Vázquez et al.), World Scientific, Singapore, (1996), 39-67.

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References:

[1]	A. Almeida and P. Hästö, Interpolation in variable exponent spaces, Rev. Mat. Complut., in press. doi: 10.1007/s13163-013-0135-1.
[2]	E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. doi: 10.1007/s00205-002-0208-7.
[3]	R. Artino and J. Barros-Neto, Hypoelliptic Boundary-value Problems, Lectures Notes in Pure and Applied Mathematics-Vol.53, Marcel Dekker, New York-Basel, 1980.
[4]	J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der mathematischen Wissenschaften-Vol.223, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[5]	P. Cerejeiras and N. Vieira, Regularization of the non-stationary Schrödinger operator, Math. Meth. in Appl. Sc., 32 (2009), 535-555. doi: 10.1002/mma.1050.
[6]	P. Cerejeiras and N. Vieira, Factorization of the non-stationary Schrödinger operator, Adv. Appl. Clifford Algebr., 17 (2007), 331-341. doi: 10.1007/s00006-007-0039-6.
[7]	P. Cerejeiras, U. Kähler and F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains, Math. Meth. in Appl. Sc., 28 (2005), 1715-1724. doi: 10.1002/mma.634.
[8]	Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.
[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, Kluwer Academic Publishers, Dordrecht etc., 1992. doi: 10.1007/978-94-011-2922-0.
[10]	L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolec Spaces with Variable Exponents, Springer-Verlang, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.
[11]	L. Diening, D. Lengeler and M. Ružička, The Stokes and Poisson problem in variable exponents spaces, Complex Var. Elliptic Equ., 56 (2011), 789-811. doi: 10.1080/17476933.2010.504843.
[12]	R. Fortini, D. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities, Nonlinear Anal., 70 (2009), 2917-2929. doi: 10.1016/j.na.2008.12.030.
[13]	K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Mathematical Methods in Practice, Wiley, Chichester, 1997.
[14]	L. Hormander, On the regularity of the solutions of boundary problems, Acta. Math., 99 (1958), 225-264.
[15]	R. F. Hoskins and J.S. Pinto, Theories of Generalised Functions - Distributions, Ultradistributions and other Generalised Functions, Horwood Publishing, Chichester, 2005. doi: 10.1533/9780857099488.
[16]	T. Kato, Nonlinear Schrödinger equation, in Schrödinger Operators, (eds. H. Holden and A. Jensen), Lectures Notes in Physics 345, Springer, Berlin, 1989. doi: 10.1007/3-540-51783-9_22.
[17]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
[18]	R. S. Kraußhar and N. Vieira, The Schrödinger equation on cylinders and the $n$-torus, J. Evol. Equ., 11 (2011), 215-237. doi: 10.1007/s00028-010-0089-4.
[19]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.
[20]	N. Laskin, Fractional quantum mechanics, Phy. Rev. E, 62 (2000), 3135-3145.
[21]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[22]	S. G. Mikhlin and S. Prössdorf, Singular Integral Operators, Springer-Verlag, Berlin etc., 1986. doi: 10.1007/978-3-642-61631-0.
[23]	H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950.
[24]	H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951.
[25]	M. Sanchón and J. M. Urbano, Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc., 361 (2009), 6387-6405. doi: 10.1090/S0002-9947-09-04399-2.
[26]	S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integr. Transf. Spec. F., 16 (2005), 461-482. doi: 10.1080/10652460412331320322.
[27]	W. Sprößig, On Helmotz decompositions and their generalizations-an overview, Math. Meth. in Appl. Sc., 33 (2009), 374-383. doi: 10.1002/mma.1212.
[28]	T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics-Vol.106, American Mathematical Society, 2006.
[29]	G. Velo, Mathematical Aspects of the nonlinear Schrödinger Equation, in Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrdinger systems: theory and applications (eds. L. Vázquez et al.), World Scientific, Singapore, (1996), 39-67.

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