Relaxation of multiple integral in subcritical Sobolev spaces (Q1130257)

The paper deals with the relaxation of integral functionals \(F(u)=\int_\Omega W(\nabla u) dx\) (\(\Omega\subset{\mathbb R}^N\), \(u\in W^{1,N}(\Omega,{\mathbb R}^N)\)) in the case when there is a gap between the weak \(W^{1,p}\) topology used for the relaxation (usually the choice of \(p\) is due to a \(p\)-growth condition from below on \(W\)) and the growth from above, of order at most \(q\). A typical example of this situation occurs when \(W(\xi)=f(\xi)+g(\text{det }\xi)\) where \(f\) has \(p\)-growth for some \(p\in (N-1,N)\) and \(g\) has linear growth (in this case \(q=N\)). The main result of the paper, under technical assumptions which include the previous example, states that if \(p>q-1\) the relaxed functional \({\mathcal F}_p(u)\) is greater than \(\int_\Omega QW(\nabla u) dx\) for any \(u\in W^{1,p}(\Omega,{\mathbb R}^N)\), with equality if \(| \nabla u| \in L^N(\Omega)\), where \(QW\) is the quasi convex envelope of \(W\). By a detailed analysis of an example it is shown that in general a strict inequality may hold if \(| \nabla u| \) is not in \(L^N(\Omega)\).