Abstract
In this paper we prove the L ∞-boundedness of solutions of the quasilinear elliptic equation
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.
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Winkert, P. L ∞-Estimates for nonlinear elliptic Neumann boundary value problems. Nonlinear Differ. Equ. Appl. 17, 289–302 (2010). https://doi.org/10.1007/s00030-009-0054-5
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DOI: https://doi.org/10.1007/s00030-009-0054-5