# Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration

created on 14 Dec 2002
modified by mingione on 09 May 2005

[BibTeX]

Published Paper

Inserted: 14 dec 2002
Last Updated: 9 may 2005

Journal: SIAM J. Mathematical Analysis
Volume: 36
Number: 5
Pages: 1540-1579
Year: 2005

Abstract:

We consider integral functionals of the type $F(u):=\int_{\Omega} f(x,u,Du)dx$ exhibiting a gap between the coercivity and the growth exponent: $$L{-1} Du p\leq f(x,u,Du)\leq L(1+ Du q)\qquad 1<p< q\,.$$We give lower semicontinuity results and conditions ensuring that the relaxed functional $\bar{F}$ is exactly $\int_{\Omega} Qf(x,u,Du)dx$, $Qf$ denoting the usual quasi-convex envelope; our conditions are sharp. Indeed we also provide counterexamples where such an integral representation fails, showing that energy concentrations appear in the relaxation procedure leading a measure representation of $\bar{F}$ with a non zero singular part, which is explicitly computed. The main point in our analysis is that such relaxation results depend in subtle way on the interaction between the ratio $q/p$ and the degree of regularity of the integrand $f$ with respect to the variable $x$. Our results extend to certain anisotropic settings, previous theorems for non-convex integrals due to Fonseca & Malý and Kristensen.

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