American Institute of Mathematical Sciences

Advanced
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 (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.   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.   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.  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, 126 (2001), 265.   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.  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.  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.  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.  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.   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.   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.  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, 39 (1992), 279.   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.   Google Scholar

[16]

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

show all references

References:
[1]

R. A. Adams and J. J. Fournier, "Sobolev Spaces," Second edition,, Pure and Applied Mathematics (Amsterdam), 140 (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.   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.   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.  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, 126 (2001), 265.   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.  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.  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.  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.  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.   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.   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.  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, 39 (1992), 279.   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.   Google Scholar

[16]

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

[1]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[2]

Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751
[3]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[4]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581
[5]

Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353
[6]

Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427
[7]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957
[8]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[9]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425
[10]

Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1205-1235. doi: 10.3934/dcds.2019052
[11]

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
[12]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[13]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[14]

Niclas Bernhoff. On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinetic & Related Models, 2010, 3 (2) : 195-222. doi: 10.3934/krm.2010.3.195
[15]

Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663
[16]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[17]

Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[18]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365
[19]

Chenchen Mou. Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2335-2362. doi: 10.3934/cpaa.2015.14.2335
[20]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences