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Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension
Error estimates for a Neumann problem in highly oscillating thin domains
1. | Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, São Paulo, SP, Brazil |
2. | Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro, SP, Brazil |
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
Y. Amirat, O. Bodart, U. de Maio and A. Gaudiello, Asymptotic Approximation of the solution of the Laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), 1598-1616.
doi: 10.1137/S0036141003414877. |
[3] |
J. M. Arrieta, "Spectral Properties of Schrödinger Operators Under Perturbations of the Domain,'' Ph. D. thesis, Georgia Inst. of Tech., 1991. |
[4] |
J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Analysis: Theory Methods and Appl., 74 (2011), 5111-5132. |
[5] |
J. M. Arrieta and M. C. Pereira, Elliptic problems in thin domains with highly oscillating boundaries, Bol. Soc. Esp. Mat. Apl., 51 (2010), 17-25. |
[6] |
J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures et Appl., 96 (2011), 29-57.
doi: 10.1016/j.matpur.2011.02.003. |
[7] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, 1978. |
[8] |
R. Brizzi and J. P. Chalot, Boundary homogenization and Neumann boundary problem, Ricerce di Matematica XLVI, 2 (1997), 341-387. |
[9] |
D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. |
[10] |
D. Cioranescu and P. Donato, "An Introduction to Homogenization,'' Oxford lecture series in mathematics and its applications, 1999. |
[11] |
D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math Anal. Appl., 71 (1979), 590-607.
doi: 10.1016/0022-247X(79)90211-7. |
[12] |
D. Cioranescu and J. S. J. Paulin, "Homogenization of Reticulated Structures,'' Springer-Verlag, 1980. |
[13] |
A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete and Continuous Dynamical Systems, 23 (2009), 197-219. |
[14] |
T. Elsken, Continuity of attractors for net-shaped thin domain, Topol. Meth. Nonlinear Analysis, 26 (2005), 315-354. |
[15] |
J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 9 (1992), 33-95. |
[16] |
J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, Rocky Mountain J. Math., 10 (1998), 125-140. |
[17] |
D. N. Arnold and A. L. Madureira, Asymptotic estimates of hierarchical, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1325-1350. |
[18] |
A. L. Madureira and F. Valentin, Asymptotics of the Poisson Problem in domains with curved rough boundaries, SIAM Journal on Mathematical Analysis, 38 (2007), 1450-1473. |
[19] |
T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwendungen, 18 (1999), 953-975. |
[20] |
J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997) 1660-1686. |
[21] |
G. Panasenko, "Multi-scale Modelling for Structures and Composites,'' Springer-Verlag, Dordrecht, 2005. |
[22] |
M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, Journal of Diff. Equations, 173 (2001), 271-320. |
[23] |
M. Prizzi and M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia mathematica, 151 (2002), 109-140. |
[24] |
G. Raugel, "Dynamics of Partial Differential Equations on Thin Domains," Lecture Notes in Math., Springer-Verlag, 1609, 1995. |
[25] |
E. Sánchez-Palencia, "Non-Homogeneous Media and Vibration Theory,'' Lecture Notes in Phys., Springer-Verlag, 127, 1980. |
[26] |
R. P. Silva, "Semicontinuidade Inferior de Atratores Para Problemas Parabólicos em Domínios Finos,'' Phd Thesis, Universidade de São Paulo, 2007. |
[27] |
L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,'' Lecture Notes of the Un. Mat. Italiana, Springer-Verlag, Berlin, 7, 2009. |
show all references
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
Y. Amirat, O. Bodart, U. de Maio and A. Gaudiello, Asymptotic Approximation of the solution of the Laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), 1598-1616.
doi: 10.1137/S0036141003414877. |
[3] |
J. M. Arrieta, "Spectral Properties of Schrödinger Operators Under Perturbations of the Domain,'' Ph. D. thesis, Georgia Inst. of Tech., 1991. |
[4] |
J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Analysis: Theory Methods and Appl., 74 (2011), 5111-5132. |
[5] |
J. M. Arrieta and M. C. Pereira, Elliptic problems in thin domains with highly oscillating boundaries, Bol. Soc. Esp. Mat. Apl., 51 (2010), 17-25. |
[6] |
J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures et Appl., 96 (2011), 29-57.
doi: 10.1016/j.matpur.2011.02.003. |
[7] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, 1978. |
[8] |
R. Brizzi and J. P. Chalot, Boundary homogenization and Neumann boundary problem, Ricerce di Matematica XLVI, 2 (1997), 341-387. |
[9] |
D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. |
[10] |
D. Cioranescu and P. Donato, "An Introduction to Homogenization,'' Oxford lecture series in mathematics and its applications, 1999. |
[11] |
D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math Anal. Appl., 71 (1979), 590-607.
doi: 10.1016/0022-247X(79)90211-7. |
[12] |
D. Cioranescu and J. S. J. Paulin, "Homogenization of Reticulated Structures,'' Springer-Verlag, 1980. |
[13] |
A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete and Continuous Dynamical Systems, 23 (2009), 197-219. |
[14] |
T. Elsken, Continuity of attractors for net-shaped thin domain, Topol. Meth. Nonlinear Analysis, 26 (2005), 315-354. |
[15] |
J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 9 (1992), 33-95. |
[16] |
J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, Rocky Mountain J. Math., 10 (1998), 125-140. |
[17] |
D. N. Arnold and A. L. Madureira, Asymptotic estimates of hierarchical, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1325-1350. |
[18] |
A. L. Madureira and F. Valentin, Asymptotics of the Poisson Problem in domains with curved rough boundaries, SIAM Journal on Mathematical Analysis, 38 (2007), 1450-1473. |
[19] |
T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwendungen, 18 (1999), 953-975. |
[20] |
J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997) 1660-1686. |
[21] |
G. Panasenko, "Multi-scale Modelling for Structures and Composites,'' Springer-Verlag, Dordrecht, 2005. |
[22] |
M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, Journal of Diff. Equations, 173 (2001), 271-320. |
[23] |
M. Prizzi and M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia mathematica, 151 (2002), 109-140. |
[24] |
G. Raugel, "Dynamics of Partial Differential Equations on Thin Domains," Lecture Notes in Math., Springer-Verlag, 1609, 1995. |
[25] |
E. Sánchez-Palencia, "Non-Homogeneous Media and Vibration Theory,'' Lecture Notes in Phys., Springer-Verlag, 127, 1980. |
[26] |
R. P. Silva, "Semicontinuidade Inferior de Atratores Para Problemas Parabólicos em Domínios Finos,'' Phd Thesis, Universidade de São Paulo, 2007. |
[27] |
L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,'' Lecture Notes of the Un. Mat. Italiana, Springer-Verlag, Berlin, 7, 2009. |
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