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Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case
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 |
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. |
show all references
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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