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

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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

Abstract
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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 (2003).
[2]	C. Amrouche, The Neumann problem in the half-space,, Comptes Rendus de l'Académie des Sciences de Paris, 335 (2002), 151.
[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.
[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. doi: 10.1016/S0021-7824(97)89945-X.
[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, 126 (2001), 265.
[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. doi: 10.1016/j.jde.2007.10.007.
[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. doi: 10.1137/090749207.
[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. doi: 10.1090/S0894-0347-02-00411-3.
[9]	J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems,, Journal of the European Mathematical Society, 9 (2007), 277. doi: 10.4171/JEMS/80.
[10]	J. Bourgain and H. Brezis, Sur l'équation div $u=f$,, Comptes Rendus de l'Académie des Sciences de Paris, 334 (2002), 973.
[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.
[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. doi: 10.1007/s00526-007-0094-9.
[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.
[14]	V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of $\mathbbR^3$,, Journal of the Faculty of Science, 39 (1992), 279.
[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.
[16]	J. Van Schaftingen, Estimates for $L^1$-vector fields,, Comptes Rendus de l'Académie des Sciences de Paris, 339 (2004), 181.

show all references

References:

[1]	R. A. Adams and J. J. Fournier, "Sobolev Spaces," Second edition,, Pure and Applied Mathematics (Amsterdam), 140 (2003).
[2]	C. Amrouche, The Neumann problem in the half-space,, Comptes Rendus de l'Académie des Sciences de Paris, 335 (2002), 151.
[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.
[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. doi: 10.1016/S0021-7824(97)89945-X.
[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, 126 (2001), 265.
[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. doi: 10.1016/j.jde.2007.10.007.
[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. doi: 10.1137/090749207.
[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. doi: 10.1090/S0894-0347-02-00411-3.
[9]	J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems,, Journal of the European Mathematical Society, 9 (2007), 277. doi: 10.4171/JEMS/80.
[10]	J. Bourgain and H. Brezis, Sur l'équation div $u=f$,, Comptes Rendus de l'Académie des Sciences de Paris, 334 (2002), 973.
[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.
[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. doi: 10.1007/s00526-007-0094-9.
[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.
[14]	V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of $\mathbbR^3$,, Journal of the Faculty of Science, 39 (1992), 279.
[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.
[16]	J. Van Schaftingen, Estimates for $L^1$-vector fields,, Comptes Rendus de l'Académie des Sciences de Paris, 339 (2004), 181.

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