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 formI(u)=∫ΩF(u,∇u),u|∂Ω=ϕ 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.