Some identities on weighted Sobolev spaces

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June 2012, 5(3): 427-434. doi: 10.3934/dcdss.2012.5.427

Some identities on weighted Sobolev spaces

Tahar Z. Boulmezaoud ^1, and Amel Kourta ^2,

1.	Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin-en-Yvelines, 45, Avenue des Etats-Unis, 78035, Versailles Cedex, France
2.	Université de Constantine, Départment de mathématiques, Route ain el bey, 25000, Constantine, Algeria

Received September 2010 Revised January 2011 Published October 2011

Abstract
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In this paper we compare some families of weigthted Sobolev spaces which are commonly used for solving partial differential equations in unbounded domains. The first result is an identity between two particular spaces. The second result is another identity which generalises partially the first one.

Keywords: unbounded domains, Weighted Sobolev spaces, interpolation., compactness.

Mathematics Subject Classification: 46A99, 46N20, 46B50, 46B7.

Citation: 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

References:

[1]	C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for Laplace's equation in $R^n$,, J. Math. Pures Appl. (9), 73 (1994), 579.
[2]	T. Z. Boulmezaoud and M. Medjden, Vorticity-vector potential formulations of the Stokes equations in the half-space,, Mathematical Methods in the Applied Sciences, 28 (2005), 903. doi: 10.1002/mma.596.
[3]	T. Z. Boulmezaoud, Espaces de Sobolev avec poids pour l'équation de Laplace dans le demi-espace,, Comptes Rendus de l'Académie des Sciences Série I Mathématiques, 328 (1999), 221.
[4]	T. Z. Boulmezaoud, On the Laplace operator and on the vector potential problems in the half-space: An approach using weighted spaces,, Mathematical Methods in the Applied Sciences, 26 (2003), 633. doi: 10.1002/mma.369.
[5]	T. Z. Boulmezaoud and U. Razafison, On the steady Oseen problem in the whole space,, Hiroshima Mathematical Journal, 35 (2005), 371.
[6]	R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in $L^q$-spaces,, In, 1530 (1992), 97.
[7]	J. Giroire, "Etude de Quelques Problèmes aux Limites Extérieurs et Résolution par Équations Intégrales",, Thèse de Doctorat d'Etat, (1987).
[8]	V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of $R^3$,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39 (1992), 279.
[9]	B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace,, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227.
[10]	J. G. Heywood, Classical solutions of the Navier-Stokes equations,, In, 771 (1980), 235.
[11]	V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points,, Trudy Moskov. Mat. Obšč., 16 (1967), 209.
[12]	A. Kufner, "Weighted Sobolev Spaces,", A Wiley-Interscience Publication, (1985).
[13]	V. G. Maz'ja and B. A. Plamenevskiĭ, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points,, In, (1977), 161.

show all references

References:

[1]	C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for Laplace's equation in $R^n$,, J. Math. Pures Appl. (9), 73 (1994), 579.
[2]	T. Z. Boulmezaoud and M. Medjden, Vorticity-vector potential formulations of the Stokes equations in the half-space,, Mathematical Methods in the Applied Sciences, 28 (2005), 903. doi: 10.1002/mma.596.
[3]	T. Z. Boulmezaoud, Espaces de Sobolev avec poids pour l'équation de Laplace dans le demi-espace,, Comptes Rendus de l'Académie des Sciences Série I Mathématiques, 328 (1999), 221.
[4]	T. Z. Boulmezaoud, On the Laplace operator and on the vector potential problems in the half-space: An approach using weighted spaces,, Mathematical Methods in the Applied Sciences, 26 (2003), 633. doi: 10.1002/mma.369.
[5]	T. Z. Boulmezaoud and U. Razafison, On the steady Oseen problem in the whole space,, Hiroshima Mathematical Journal, 35 (2005), 371.
[6]	R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in $L^q$-spaces,, In, 1530 (1992), 97.
[7]	J. Giroire, "Etude de Quelques Problèmes aux Limites Extérieurs et Résolution par Équations Intégrales",, Thèse de Doctorat d'Etat, (1987).
[8]	V. Girault, The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of $R^3$,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39 (1992), 279.
[9]	B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace,, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227.
[10]	J. G. Heywood, Classical solutions of the Navier-Stokes equations,, In, 771 (1980), 235.
[11]	V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points,, Trudy Moskov. Mat. Obšč., 16 (1967), 209.
[12]	A. Kufner, "Weighted Sobolev Spaces,", A Wiley-Interscience Publication, (1985).
[13]	V. G. Maz'ja and B. A. Plamenevskiĭ, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points,, In, (1977), 161.

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