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L -Estimates for nonlinear elliptic Neumann boundary value problems

  • Patrick Winkert
Article

Abstract

In this paper we prove the L -boundedness of solutions of the quasilinear elliptic equation
$$ \begin{array}{ll} Au \, = f(x,u,\nabla u) &\quad \rm{in }\, \Omega, \\ \dfrac{\partial u}{ \partial \nu} \, = g(x,u) &\quad \rm{on }\, \partial \Omega, \end{array} $$
where A is a second order quasilinear differential operator and \({f:\Omega \times \mathbb{R} \times \mathbb{R}^N \rightarrow \mathbb{R}}\) as well as \({g: \partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}}\) are Carathéodory functions satisfying natural growth conditions. Our main result is given in Theorem 4.1 and is based on the Moser iteration technique along with real interpolation theory. Furthermore, the solutions of the elliptic equation above belong to \({C^{1,\alpha}(\overline{\Omega})}\), if g is Hölder continuous.

Mathematics Subject Classification (2000)

35B45 35J25 35J60 

Keywords

A priori estimates Neumann boundary values Nonlinear elliptic equations 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Department of MathematicsMartin-Luther-University Halle-WittenbergHalleGermany

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