• Previous Article
    On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity
  • CPAA Home
  • This Issue
  • Next Article
    Existence and uniqueness for $\mathbb{D}$-solutions of reflected BSDEs with two barriers without Mokobodzki's condition
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, Academic Press, 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, Comm. Pure Appl. Math, 12 (1959), 623-727.  Google Scholar

[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.  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, Ann. Scuola. Norm. Sup. Pisa Cl. Sci, 22 (1995), 241-273.  Google Scholar

[5]

M. Chipot, Elliptic Equations, An Introductory Cours, Birkhauser, ISBN: 978-3764399818, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

[6]

M. Chipot, On some anisotropic singular perturbation problems, Asymptotic Analysis, 55 (2007), 125-144.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Mathematica, 130 (1973), 309-317.  Google Scholar

[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.  Google Scholar

[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.  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, Ann. Sc. Norm. Sup. Pisa, 18 (1964), 385-387.  Google Scholar

show all references

References:
[1]

R.A. Adams and John J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, 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, Comm. Pure Appl. Math, 12 (1959), 623-727.  Google Scholar

[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.  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, Ann. Scuola. Norm. Sup. Pisa Cl. Sci, 22 (1995), 241-273.  Google Scholar

[5]

M. Chipot, Elliptic Equations, An Introductory Cours, Birkhauser, ISBN: 978-3764399818, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

[6]

M. Chipot, On some anisotropic singular perturbation problems, Asymptotic Analysis, 55 (2007), 125-144.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Mathematica, 130 (1973), 309-317.  Google Scholar

[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.  Google Scholar

[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.  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, Ann. Sc. Norm. Sup. Pisa, 18 (1964), 385-387.  Google Scholar

[1]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179

[2]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008

[3]

Hugo Beirão da Veiga. Turbulence models, $p-$fluid flows, and $W^{2, L}$ regularity of solutions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 769-783. doi: 10.3934/cpaa.2009.8.769

[4]

Huijiang Zhao, Yinchuan Zhao. Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1243-1262. doi: 10.3934/dcds.2003.9.1243

[5]

Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252

[6]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61

[7]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431

[8]

Xianling Fan, Yuanzhang Zhao, Guifang Huang. Existence of solutions for the $p-$Laplacian with crossing nonlinearity. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1019-1024. doi: 10.3934/dcds.2002.8.1019

[9]

Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021

[10]

Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793

[11]

Maciej Smołka. Asymptotic behaviour of optimal solutions of control problems governed by inclusions. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 641-652. doi: 10.3934/dcds.1998.4.641

[12]

Riccardo Molle, Donato Passaseo. On the behaviour of the solutions for a class of nonlinear elliptic problems in exterior domains. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 445-454. doi: 10.3934/dcds.1998.4.445

[13]

Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167

[14]

Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321

[15]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363

[16]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[17]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761

[18]

Brahim Alouini. Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021032

[19]

Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217

[20]

Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 5019-5045. doi: 10.3934/dcds.2020210

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]