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

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

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 $L^1$.
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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