December  2007, 6(4): 957-982. doi: 10.3934/cpaa.2007.6.957

Singular boundary conditions and regularity for the biharmonic problem 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, France, France

Received  August 2006 Revised  May 2007 Published  September 2007

In this paper, we are interested in some aspects of the biharmonic equation in the half-space $\mathbb R^N_+$, with $N\geq 2$. We study the regularity of generalized solutions in weighted Sobolev spaces, then we consider the question of singular boundary conditions. To finish, we envisage other sorts of boundary conditions.
Citation: 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
[1]

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

[2]

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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Linglong Du. Characteristic half space problem for the Broadwell model. Networks & Heterogeneous Media, 2014, 9 (1) : 97-110. doi: 10.3934/nhm.2014.9.97

[14]

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

[15]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148

[16]

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

[17]

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

[18]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061

[19]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167

[20]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]