# American Institute of Mathematical Sciences

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

 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.
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:
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##### 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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