Calculus of Variations and Geometric Measure Theory

P. Bousquet - C. Mariconda - G. Treu

On the Lavrentiev phenomenon for multiple integral scalar variational problems

created by mariconda on 10 Jun 2013

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Inserted: 10 jun 2013
Last Updated: 10 jun 2013

Year: 2013

Abstract:

Let $\phi$ be a Lipschitz map on $\mathbb R^n$. We prove the non occurrence of the Lavrentiev gap between Lipschitz functions and Sobolev functions for functionals of the form \[I(u)=\int_{\Omega}F(u,\nabla u)\quad u\in \phi+W^{1,p}_{0}(\Omega) \quad (1\le p<+\infty)\] when $\Omega$ belongs to a wide class of open and bounded subsets of $\mathbb R^n$ containing Lipschitz ones, and either $F$ is convex in both variables or $F(s,\xi)=a(s)g(\xi)+b(s)$ with $g$ convex and $s\mapsto a(s)g(0)+b(s)$ satisfying a non oscillatory condition at infinity. We derive the non occurrence of the Lavrentiev phenomenon for unnecessarily convex functionals of the gradient. No growth conditions are assumed.


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