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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), 13-24.  Google Scholar

[2]

B. E, J. Dahlberg, Harmonic functions in Lipschitz domains, Proceedinds of Symposia in Pure Mathematics, XXXV, part 1 (1979), 319-322.  Google Scholar

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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), 297-314.  Google Scholar

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D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, New York, 1983.  Google Scholar

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

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E. M. Stein, "Singular Integrals and differentiability Properties of Functions," Princeton University Press, Princeton, 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), 13-24.  Google Scholar

[2]

B. E, J. Dahlberg, Harmonic functions in Lipschitz domains, Proceedinds of Symposia in Pure Mathematics, XXXV, part 1 (1979), 319-322.  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), 297-314.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, New York, 1983.  Google Scholar

[5]

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

[6]

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

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