Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians

Abstract

We consider the functional $$F_\infty(u)=\int_Ωf(x,u(x),\nabla u(x)) dx \quad\quad u\in φ+ W_0^{1,\infty}(Ω,\mathbb{R})$$ where $Ω$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $φ\in W^{1,\infty}(Ω)$. We do not assume neither convexity or continuity of the Lagrangian w.r.t. the last variable. We prove that, under suitable assumptions, the lower semicontinuous envelope of $F_\infty$ both in $φ+W^{1,\infty}(Ω)$ and in the larger space $φ+W^{1,p}(Ω)$ can be represented by means of the bipolar $f^{**}$ of $f$. In particular we can also exclude Lavrentiev Phenomenon between $W^{1,\infty}(Ω)$ and $W^{1,1}(Ω)$ for autonomous Lagrangians.