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

Some identities on weighted Sobolev spaces

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.
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

[1]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[2]

T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105

[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]

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, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011

[5]

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

[6]

Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901

[7]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

[8]

Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021147

[9]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[10]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1

[11]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241

[12]

Esa V. Vesalainen. Rellich type theorems for unbounded domains. Inverse Problems & Imaging, 2014, 8 (3) : 865-883. doi: 10.3934/ipi.2014.8.865

[13]

Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74

[14]

Sergiĭ Kolyada. A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1291-1317. doi: 10.3934/dcdss.2020074

[15]

Valerii Los, Vladimir Mikhailets, Aleksandr Murach. Parabolic problems in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021123

[16]

Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969

[17]

Dario Cordero-Erausquin, Alessio Figalli. Regularity of monotone transport maps between unbounded domains. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297

[18]

Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800

[19]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[20]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]