American Institute of Mathematical Sciences

Advanced
March  2011, 10(2): 561-570. doi: 10.3934/cpaa.2011.10.561

The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions

Yi Cao 1, , Dong Li 2, and  Lihe Wang 3,

1. 

College of Science, Xi'an Jiaotong University, Xi'an, 710049, China
2. 

Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242
3. 

Department of Mathematics, Shanghai Jiaotong University, Shang hai 200240, China

Received  April 2010 Revised  August 2010 Published  December 2010

In this paper we study uniformly elliptic equations with non-compatible conditions, where $\Omega$ is a bounded Lipchitz domain, and the right-hand side term and the boundary value of the elliptic equations belong to $L^p (p \geq 2)$ space. Then the optimal weighted $W^{2, p}$ estimates will be given by Whitney decomposition and $L^p$ estimates of non-tangential maximal function associated to solutions of the elliptic equations.
Keywords: non-compatible conditions., Elliptic equations, bounded Lipchitz domain.
Mathematics Subject Classification: Primary: 35J17, 35J60; Secondary: 47B2.
Citation: Yi Cao, Dong Li, Lihe Wang. The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions. Communications on Pure & Applied Analysis, 2011, 10 (2) : 561-570. doi: 10.3934/cpaa.2011.10.561
References:
[1]

B. E, J. Dahlberg, On the poisson integral for Lipschitz and $C^1$ domains,, Studia Math., 66 (1979), 13.   Google Scholar

[2]

B. E, J. Dahlberg, Harmonic functions in Lipschitz domains,, Proceedinds of Symposia in Pure Mathematics, XXXV, part 1 (1979), 319.   Google Scholar

[3]

B. E, J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the non-tangential maximal functions for harmonic function in a Lipschitz domain,, Studia Math., 67 (1980), 297.   Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983).   Google Scholar

[5]

C. E. Kenig and Cora Sadosky, "Harmonic Analysis and Partial Differential Equations,", Chicago Lectures in Mathematics, (1999).   Google Scholar

[6]

E. M. Stein, "Singular Integrals and differentiability Properties of Functions,", Princeton University Press, (1970).   Google Scholar

show all references

References:
[1]

B. E, J. Dahlberg, On the poisson integral for Lipschitz and $C^1$ domains,, Studia Math., 66 (1979), 13.   Google Scholar

[2]

B. E, J. Dahlberg, Harmonic functions in Lipschitz domains,, Proceedinds of Symposia in Pure Mathematics, XXXV, part 1 (1979), 319.   Google Scholar

[3]

B. E, J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the non-tangential maximal functions for harmonic function in a Lipschitz domain,, Studia Math., 67 (1980), 297.   Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983).   Google Scholar

[5]

C. E. Kenig and Cora Sadosky, "Harmonic Analysis and Partial Differential Equations,", Chicago Lectures in Mathematics, (1999).   Google Scholar

[6]

E. M. Stein, "Singular Integrals and differentiability Properties of Functions,", Princeton University Press, (1970).   Google Scholar

[1]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[2]

N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079
[3]

Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019209
[4]

Manuel Núñez. Existence of solutions of the equations of electron magnetohydrodynamics in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1019-1034. doi: 10.3934/dcds.2010.26.1019
[5]

Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431
[6]

Adnan Ben Aziza, Mohamed Ben Chrouda. Characterization for the existence of bounded solutions to elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1157-1170. doi: 10.3934/dcds.2019049
[7]

Zaynab Salloum. Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 625-642. doi: 10.3934/cpaa.2010.9.625
[8]

Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783
[9]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047
[10]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991
[11]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[12]

Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387
[13]

Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic & Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002
[14]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[15]

Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333
[16]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057
[17]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135
[18]

Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520
[19]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[20]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences