W1,p estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains

By Jun Geng. 2012

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Abstract We study W 1 , p estimates in Lipschitz domains for second order elliptic equations and systems of divergence form with real-valued, bounded, measurable coefficients. For any fixed p > 2 , we prove that a weak reverse Holder inequality implies the W 1 , p estimates for solutions with Neumann boundary conditions. As a result, we are able to show that if the coefficient matrix of elliptic equation is symmetric and in VMO ( R n ) , the W 1 , p estimate holds for 3 2 − e p 3 + e if n ≥ 3 , and for 4 3 − e p 4 + e if n = 2 , and for elliptic systems the W 1 , p estimate is true for 2 n n + 1 − e p 2 n n − 1 + e .

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/cf3858dcd4f67adeb1fb6c73bef591b4230ded83