Abstract
In a recent paper we proved the non occurrence of the Lavrentiev gap between Lipschitz and Sobolev functions for functionals of the form
when \(\phi :\mathbb {R}^{n} \rightarrow \mathbb {R}\) is Lipschitz, Ω belongs to a wide class of open bounded sets in \(\mathbb {R}^{n}\) containing Lipschitz domains, and the lagrangian F is assumed to be either convex in both variables or a sum of functions F(s, ξ) = a(s)g(ξ) + b(s) with g convex and s ↦ a(s)g(0) + b(s) satisfying a non oscillatory condition at infinity. In this survey we discuss the state of the art on the subject and give a self-contained proof of our result in the simpler case of a (strongly) star-shaped domain, for a lagrangian depending just on the gradient; in particular we point out what are the main difficulties to overcome in order to get the result without assuming growth conditions. We also formulate, and prove, a characterization of a useful class of star-shaped domains in terms of the radii function.
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Bousquet, P., Mariconda, C. & Treu, G. A Survey on the Non Occurence of the Lavrentiev Gap for Convex, Autonomous Multiple Integral Scalar Variational Problems. Set-Valued Var. Anal 23, 55–68 (2015). https://doi.org/10.1007/s11228-014-0305-4
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DOI: https://doi.org/10.1007/s11228-014-0305-4