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Continuity of solutions of a nonlinear elliptic equation

Published online by Cambridge University Press:  16 January 2012

Pierre Bousquet
Pierre Bousquet*
Affiliation:
Université Aix-Marseille 1, LATP UMR6632 3 place Victor Hugo, 13331 Marseille Cedex 3, France. bousquet@cmi.univ-mrs.fr
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Abstract

We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when a is radial: \hbox{$a(\xi)=\fr{l(|\xi|)}{|\xi|} \xi$}a(ξ)=l(|ξ|)|ξ|ξ for some increasing l:ℝ+ → ℝ+.

Keywords

Nonlinear elliptic equationscontinuity of solutionslower bounded slope conditionLavrentiev phenomenon
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 1 - 19
Copyright
© EDP Sciences, SMAI, 2012

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