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lavrentiev phenomenon

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Continuity of solutions of a nonlinear ellipticequation
Pierre Bousquet
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 19 / Issue 1 / January 2013

Published online by Cambridge University Press:

16 January 2012, pp. 1-19

Print publication:

January 2013
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We consider a nonlinear elliptic equation of the formdiv [a(∇u)] + F[u] = 0on a domain Ω, subject to a Dirichlet boundary conditiontru = φ. We do not assume that the higher order terma satisfies growth conditions from above. We prove the existence ofcontinuous solutions either when Ω is convex and φ satisfies a one-sidedbounded slope condition, or when a is radial:\hbox{$a(\xi)=\fr{l(|\xi|)}{|\xi|} \xi$} $a (ξ) = \frac{l (| ξ |)}{| ξ |} ξ$ for some increasingl:ℝ+ → ℝ+.