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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

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), 579-606.  Google Scholar

[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), 903-915. doi: 10.1002/mma.596.  Google Scholar

[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), 221-226.  Google Scholar

[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), 633-669. doi: 10.1002/mma.369.  Google Scholar

[5]

T. Z. Boulmezaoud and U. Razafison, On the steady Oseen problem in the whole space, Hiroshima Mathematical Journal, 35 (2005), 371-401.  Google Scholar

[6]

R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in $L^q$-spaces, In "The Navier-Stokes Equations II-Theory and Numerical Methods" (Oberwolfach, 1991), Lecture Notes in Math., 1530, Springer, Berlin, (1992), 97-110.  Google Scholar

[7]

J. Giroire, "Etude de Quelques Problèmes aux Limites Extérieurs et Résolution par Équations Intégrales", Thèse de Doctorat d'Etat, Université Pierre et Marie Curie, Paris, 1987. Google Scholar

[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), 279-307.  Google Scholar

[9]

B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227-272.  Google Scholar

[10]

J. G. Heywood, Classical solutions of the Navier-Stokes equations, In "Approximation Methods for Navier-Stokes Problems" (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), Lecture Notes in Math., 771, Springer, Berlin-New York, (1980), 235-248.  Google Scholar

[11]

V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16 (1967), 209-292.  Google Scholar

[12]

A. Kufner, "Weighted Sobolev Spaces," A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985.  Google Scholar

[13]

V. G. Maz'ja and B. A. Plamenevskiĭ, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points, In "Elliptische Differentialgleichungen" (Meeting, Rostock, 1977), 161-190, Wilhelm-Pieck-Univ., Rostock, 1978.  Google Scholar

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), 579-606.  Google Scholar

[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), 903-915. doi: 10.1002/mma.596.  Google Scholar

[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), 221-226.  Google Scholar

[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), 633-669. doi: 10.1002/mma.369.  Google Scholar

[5]

T. Z. Boulmezaoud and U. Razafison, On the steady Oseen problem in the whole space, Hiroshima Mathematical Journal, 35 (2005), 371-401.  Google Scholar

[6]

R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in $L^q$-spaces, In "The Navier-Stokes Equations II-Theory and Numerical Methods" (Oberwolfach, 1991), Lecture Notes in Math., 1530, Springer, Berlin, (1992), 97-110.  Google Scholar

[7]

J. Giroire, "Etude de Quelques Problèmes aux Limites Extérieurs et Résolution par Équations Intégrales", Thèse de Doctorat d'Etat, Université Pierre et Marie Curie, Paris, 1987. Google Scholar

[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), 279-307.  Google Scholar

[9]

B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227-272.  Google Scholar

[10]

J. G. Heywood, Classical solutions of the Navier-Stokes equations, In "Approximation Methods for Navier-Stokes Problems" (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), Lecture Notes in Math., 771, Springer, Berlin-New York, (1980), 235-248.  Google Scholar

[11]

V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16 (1967), 209-292.  Google Scholar

[12]

A. Kufner, "Weighted Sobolev Spaces," A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985.  Google Scholar

[13]

V. G. Maz'ja and B. A. Plamenevskiĭ, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points, In "Elliptische Differentialgleichungen" (Meeting, Rostock, 1977), 161-190, Wilhelm-Pieck-Univ., Rostock, 1978.  Google Scholar

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