On the Lavrentiev phenomenon for multiple integral scalar variational problems

Abstract

We prove the non-occurrence of Lavrentiev gaps between Lipschitz and Sobolev functions for functionals of the form, when φ:Rn → R is Lipschitz and Ω belongs to a wide class of open bounded sets in Rn containing Lipschitz domains. 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. We thus derive the non-occurrence of the Lavrentiev phenomenon for unnecessarily convex functionals of the gradient. No growth conditions are assumed. © 2013 Elsevier Inc.