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Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations

Pierre Bousquet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form div a ( u ) + F [ u ] ( x ) = 0 , over the functions u W 1 , 1 ( Ω ) that assume given boundary values ϕ on ∂Ω. The vector field a : n n satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...

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