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Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
AbstractWe consider the classical functional of the Calculus of Variations of the formwhere Ω is a bounded open subset of ân and F : Ω Ă â Ă ân â â is a CarathĂ©odory convex function; the admissible functions u coincide with a prescribed Lipschitz function Ï on âΩ. We formulate some conditions under which a given function in Ï + (Ω) with I(u) < +â can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with Ï on âΩ. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, Ο) = f(x, u) + h(x, Ο) is convex and x ⊠F(x, 0, 0) is sufficiently smooth