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

TL;DR

This work addresses the relaxation of integral functionals with nonconvex Lagrangians by proving that the sequential weak lower semicontinuous envelope of $F_q(u)$ is represented by the convex envelope $f^{**}$ in the gradient variable, i.e., $\mathrm{sc}^-(F_p)(u)=\int_{\Omega} f^{**}(x,u,\nabla u) dx$ under suitable hypotheses. The authors develop explicit recovery sequences using a Vitali covering argument for nonautonomous Lagrangians and truncation techniques for the general case, enabling a constructive relaxation framework even without gradient continuity. They establish a Lavrentiev gap-transfer principle: absence of a Lavrentiev gap for the relaxed functional implies absence for the original, linking relaxation to numerical stability. In the autonomous case, they combine the general results with recent approximations by Lipschitz functions (Bousquet 2023) to obtain strong approximation properties and to exclude Lavrentiev phenomena under broader, natural hypotheses, with clear implications for finite-element-type discretizations.

Abstract

We consider the functional $$F_\infty(u)=\int_Ωf(x,u(x),\nabla u(x)) dx \quad\quad u\in \varphi+ W_0^{1,\infty}(Ω,\mathbb{R})$$ where $Ω$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $\varphi\in W^{1,\infty}(Ω)$. 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 $\varphi+W^{1,\infty}(Ω)$ and in the larger space $\varphi+W^{1,p}(Ω)$ 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.

Theorems & Definitions (43)

• Remark 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Remark 6
• Proposition 8
• proof