On the L^p- theory of Anisotropic singular perturbations of elliptic problems

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July 2016, 15(4): 1157-1178. doi: 10.3934/cpaa.2016.15.1157

On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems

Ogabi Chokri ^1,

1.	Académie de Grenoble, Lycée Saint-Marc, Nivolas-Vermelle, 38300, France

Received April 2015 Revised January 2016 Published April 2016

Abstract
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In this article we give an extension of the $L^2-$theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving $L^{p}$ data ($1 < p < 2$). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains.

Keywords: entropy solutions, $L^p-$ theory, rate of convergence., Anisotropic singular perturbations, elliptic problems, Asymptotic behaviour.

Mathematics Subject Classification: 35J15, 35B2.

Citation: Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157

References:

[1]	R.A. Adams and John J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, 2003.
[2]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math, 12 (1959), 623-727.
[3]	L. Boccardo, T. Gallouët and J.L. Vazquez, Nonlinear elliptic equations in $R^n$ without growth restrictions on the data, Journal of Differential Equations, 105 (1993), 334-363. doi: 10.1006/jdeq.1993.1092.
[4]	Ph. Bénilan, Philippe, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and J.L. Vazquez, An $L^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola. Norm. Sup. Pisa Cl. Sci, 22 (1995), 241-273.
[5]	M. Chipot, Elliptic Equations, An Introductory Cours, Birkhauser, ISBN: 978-3764399818, 2009. doi: 10.1007/978-3-7643-9982-5.
[6]	M. Chipot, On some anisotropic singular perturbation problems, Asymptotic Analysis, 55 (2007), 125-144.
[7]	M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 21-26. doi: 10.1016/j.crma.2007.12.004.
[8]	M. Chipot and S. Guesmia, On the asymptotic behaviour of elliptic, anisotropic singular perturbations problems, Com. Pur. App. Ana, 8 (2009), 179-193. doi: 10.3934/cpaa.2009.8.179.
[9]	M. Chipot, S. Guesmia and M. Sengouga, Singular perturbations of some nonlinear problems, J. Math. Sci, 176 2011, 828-843. doi: 10.1007/s10958-011-0439-y.
[10]	M. Chipot and S. Guesmia, On a class of integro-differential problems, Commun. Pure Appl. Anal., 9 2010, 1249-1262. doi: 10.3934/cpaa.2010.9.1249.
[11]	P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Mathematica, 130 (1973), 309-317.
[12]	S. Fucik, O. John and J. Necas, On the existence of Schauder basis in Sobolev spaces, Comment. Math. Univ. Carolin, 13 (1972), 163-175.
[13]	T. Gallouet and R. Herbin, Existence of a solution to a coupled elliptic system, Appl.Math. Letters, 7 (1994), 49-55. doi: 10.1016/0893-9659(94)90030-2.
[14]	C. Ogabi, On a class of nonlinear elliptic, anisotropic singular perturbations problems,, \textit{Preprint:} \url{https://hal.archives-ouvertes.fr/hal-01074262}., ().
[15]	J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Sup. Pisa, 18 (1964), 385-387.

show all references

References:

[1]	R.A. Adams and John J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, 2003.
[2]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math, 12 (1959), 623-727.
[3]	L. Boccardo, T. Gallouët and J.L. Vazquez, Nonlinear elliptic equations in $R^n$ without growth restrictions on the data, Journal of Differential Equations, 105 (1993), 334-363. doi: 10.1006/jdeq.1993.1092.
[4]	Ph. Bénilan, Philippe, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and J.L. Vazquez, An $L^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola. Norm. Sup. Pisa Cl. Sci, 22 (1995), 241-273.
[5]	M. Chipot, Elliptic Equations, An Introductory Cours, Birkhauser, ISBN: 978-3764399818, 2009. doi: 10.1007/978-3-7643-9982-5.
[6]	M. Chipot, On some anisotropic singular perturbation problems, Asymptotic Analysis, 55 (2007), 125-144.
[7]	M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 21-26. doi: 10.1016/j.crma.2007.12.004.
[8]	M. Chipot and S. Guesmia, On the asymptotic behaviour of elliptic, anisotropic singular perturbations problems, Com. Pur. App. Ana, 8 (2009), 179-193. doi: 10.3934/cpaa.2009.8.179.
[9]	M. Chipot, S. Guesmia and M. Sengouga, Singular perturbations of some nonlinear problems, J. Math. Sci, 176 2011, 828-843. doi: 10.1007/s10958-011-0439-y.
[10]	M. Chipot and S. Guesmia, On a class of integro-differential problems, Commun. Pure Appl. Anal., 9 2010, 1249-1262. doi: 10.3934/cpaa.2010.9.1249.
[11]	P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Mathematica, 130 (1973), 309-317.
[12]	S. Fucik, O. John and J. Necas, On the existence of Schauder basis in Sobolev spaces, Comment. Math. Univ. Carolin, 13 (1972), 163-175.
[13]	T. Gallouet and R. Herbin, Existence of a solution to a coupled elliptic system, Appl.Math. Letters, 7 (1994), 49-55. doi: 10.1016/0893-9659(94)90030-2.
[14]	C. Ogabi, On a class of nonlinear elliptic, anisotropic singular perturbations problems,, \textit{Preprint:} \url{https://hal.archives-ouvertes.fr/hal-01074262}., ().
[15]	J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Sup. Pisa, 18 (1964), 385-387.

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