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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Quasilinear elliptic equations with VMO coefficients


Author: Dian K. Palagachev
Journal: Trans. Amer. Math. Soc. 347 (1995), 2481-2493
MSC: Primary 35J65
DOI: https://doi.org/10.1090/S0002-9947-1995-1308019-6
MathSciNet review: 1308019
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Abstract: Strong solvability and uniqueness in Sobolev space $ {W^{2,n}}(\Omega )$ are proved for the Dirichlet problem

$\displaystyle \left\{ {_{u = \varphi \quad {\text{on}}\partial \Omega .}^{{a^{i... ...{D_{ij}}u + b(x,u,Du) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\Omega }} \right.$

It is assumed that the coefficients of the quasilinear elliptic operator satisfy Carathéodory's condition, the $ {a^{ij}}$ are $ V\, M\, O$ functions with respect to $ x$, and structure conditions on $ b$ are required. The main results are derived by means of the Aleksandrov-Pucci maximum principle and Leray-Schauder's fixed point theorem via a priori estimate for the $ {L^{2n}}$-norm of the gradient.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1308019-6
Keywords: Strong solution of elliptic PDE, VMO, a priori estimates, Aleksandrov-Pucci maximum principle, Leray-Schauder fixed point theorem
Article copyright: © Copyright 1995 American Mathematical Society