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June  2012, 5(3): 369-397. doi: 10.3934/dcdss.2012.5.369

Elliptic problems with $L^1$-data in the half-space

Chérif Amrouche 1, and  Huy Hoang Nguyen 2,

1. 

Laboratoire de Mathématiques Appliquées, CNRS UMR 5142, Université de Pau et des Pays de l’Adour, IPRA, Avenue de l’Université, 64000 Pau
2. 

Departamento de Matematica, IMECC - Universidade Estadual de Campinas, Caixa Postal 6065, Campinas, SP 13083-970, Brazil

Received  September 2010 Revised  February 2011 Published  October 2011

In this paper, we study the div-curl-grad operators and some elliptic problems in the half-space $\mathbb{R}^n_+$, with $n\geq 2$. We consider data in weighted Sobolev spaces and in $L^1$.
Keywords: half-space, weighted Sobolev spaces., Div-curl-grad operators, elliptic.
Mathematics Subject Classification: 35J25, 35J4.
Citation: Chérif Amrouche, Huy Hoang Nguyen. Elliptic problems with $L^1$-data in the half-space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 369-397. doi: 10.3934/dcdss.2012.5.369
References:
[1]

R. A. Adams and J. J. Fournier, "Sobolev Spaces," Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

C. Amrouche, The Neumann problem in the half-space, Comptes Rendus de l'Académie des Sciences de Paris, 335 (2002), 151-156.  Google Scholar

[3]

C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for Laplace's equation in $\R^n$, Journal de Mathématiques Pures et Appliquées (9), 73 (1994), 579-606.  Google Scholar

[4]

C. Amrouche, V. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: An approach in weighted Sobolev spaces, Journal de Mathématiques Pures et Appliquées (9), 76 (1997), 55-81. doi: 10.1016/S0021-7824(97)89945-X.  Google Scholar

[5]

C. Amrouche and Š. Nečasová, Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), Mathematica Bohemica, 126 (2001), 265-274.  Google Scholar

[6]

C. Amrouche, Š. Nečasová and Y. Raudin, Very weak, generalized and strong solutions to the Stokes system in the half-space, Journal of Differential Equations, 244 (2008), 887-915. doi: 10.1016/j.jde.2007.10.007.  Google Scholar

[7]

C. Amrouche, Š. Nečasová and Y. Raudin, From strong to very weak solutions to the Stokes system with Navier boundary conditions in the half-space, SIAM J. Math. Anal., 41 (2009), 1792-1815. doi: 10.1137/090749207.  Google Scholar

[8]

J. Bourgain and H. Brezis, On the equation div $Y=f$ and application to control of phases, Journal of the American Mathematical Society, 16 (2003), 393-426. doi: 10.1090/S0894-0347-02-00411-3.  Google Scholar

[9]

J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems, Journal of the European Mathematical Society, 9 (2007), 277-315. doi: 10.4171/JEMS/80.  Google Scholar

[10]

J. Bourgain and H. Brezis, Sur l'équation div $u=f$, Comptes Rendus de l'Académie des Sciences de Paris, 334 (2002), 973-976.  Google Scholar

[11]

J. Bourgain and H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, Comptes Rendus de l'Académie des Sciences de Paris, 338 (2004), 539-543.  Google Scholar

[12]

H. Brezis and J. Van Schaftingen, Boundary extimates for elliptic systems with $\mathbfL^1$-data, Calculus of Variations and Partial Differential Equations, 30 (2007), 369-388. doi: 10.1007/s00526-007-0094-9.  Google Scholar

[13]

J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Annales de l'Institut Fourier, 5 (): 1953.  doi: 10.5802/aif.55.  Google Scholar

[14]

V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of $\mathbbR^3$, Journal of the Faculty of Science, The University of Tokyo, Sect. IA Math., 39 (1992), 279-307.  Google Scholar

[15]

B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rendiconti del Seminario Matematico della Université di Padova, 46 (1971), 227-272.  Google Scholar

[16]

J. Van Schaftingen, Estimates for $L^1$-vector fields, Comptes Rendus de l'Académie des Sciences de Paris, 339 (2004), 181-186.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. Fournier, "Sobolev Spaces," Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

C. Amrouche, The Neumann problem in the half-space, Comptes Rendus de l'Académie des Sciences de Paris, 335 (2002), 151-156.  Google Scholar

[3]

C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for Laplace's equation in $\R^n$, Journal de Mathématiques Pures et Appliquées (9), 73 (1994), 579-606.  Google Scholar

[4]

C. Amrouche, V. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: An approach in weighted Sobolev spaces, Journal de Mathématiques Pures et Appliquées (9), 76 (1997), 55-81. doi: 10.1016/S0021-7824(97)89945-X.  Google Scholar

[5]

C. Amrouche and Š. Nečasová, Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), Mathematica Bohemica, 126 (2001), 265-274.  Google Scholar

[6]

C. Amrouche, Š. Nečasová and Y. Raudin, Very weak, generalized and strong solutions to the Stokes system in the half-space, Journal of Differential Equations, 244 (2008), 887-915. doi: 10.1016/j.jde.2007.10.007.  Google Scholar

[7]

C. Amrouche, Š. Nečasová and Y. Raudin, From strong to very weak solutions to the Stokes system with Navier boundary conditions in the half-space, SIAM J. Math. Anal., 41 (2009), 1792-1815. doi: 10.1137/090749207.  Google Scholar

[8]

J. Bourgain and H. Brezis, On the equation div $Y=f$ and application to control of phases, Journal of the American Mathematical Society, 16 (2003), 393-426. doi: 10.1090/S0894-0347-02-00411-3.  Google Scholar

[9]

J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems, Journal of the European Mathematical Society, 9 (2007), 277-315. doi: 10.4171/JEMS/80.  Google Scholar

[10]

J. Bourgain and H. Brezis, Sur l'équation div $u=f$, Comptes Rendus de l'Académie des Sciences de Paris, 334 (2002), 973-976.  Google Scholar

[11]

J. Bourgain and H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, Comptes Rendus de l'Académie des Sciences de Paris, 338 (2004), 539-543.  Google Scholar

[12]

H. Brezis and J. Van Schaftingen, Boundary extimates for elliptic systems with $\mathbfL^1$-data, Calculus of Variations and Partial Differential Equations, 30 (2007), 369-388. doi: 10.1007/s00526-007-0094-9.  Google Scholar

[13]

J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Annales de l'Institut Fourier, 5 (): 1953.  doi: 10.5802/aif.55.  Google Scholar

[14]

V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of $\mathbbR^3$, Journal of the Faculty of Science, The University of Tokyo, Sect. IA Math., 39 (1992), 279-307.  Google Scholar

[15]

B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rendiconti del Seminario Matematico della Université di Padova, 46 (1971), 227-272.  Google Scholar

[16]

J. Van Schaftingen, Estimates for $L^1$-vector fields, Comptes Rendus de l'Académie des Sciences de Paris, 339 (2004), 181-186.  Google Scholar

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