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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

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

Received  April 2015 Revised  January 2016 Published  April 2016

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

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