Some remarks on the identity between a variational integral and its relaxed functional. (English) Zbl 0715.49002

The function \(f(x,z)=(x_ 2/| x|)| z\cdot x| +| z|^ p\), with \(x\in R^ 2-\{0\}\), \(z\in R^ 2\), \(1<p<2\) and the set \(B=\{x\in R^ 2:| x| <1\}\) are considered. It is proved that, in spite of the fact that \(| z|^ p\leq f(x,z)\leq L(a(x)+| z|^ q)\) for some \(q>2\), \(L\geq 1\), a in \(L^ s_{loc}(R^ 2)\) with \(s>1\) and \(| z|^ p\leq f(x,z)\leq w(x)| z|^ p\) for some w in \(L^ s_{loc}(R^ 2)\) with \(s>1\), the functional \(u\in W^{1,1}(B)\mapsto \int_{B}f(x,Du)\), that is \(L^ 1(B)\)-lower semicontinuous on \(W^{1,1}(B)\), does not agree on \(W^{1,1}(B)\) with its relaxed functional in the topology \(L^ 1(B)\) defined by inf\(\{\liminf_{h}\int_{B}f(x,Du_ h)\), \(u_ h\) Lipschitz continuous, \(u_ h\to u\) in \(L^ 1(B)\}.\)
By virtue of this example the paper yields a case of an integral functional, with integrand depending only on x and the gradient of u, for which a Lavrentieff phenomenon occurs. Sufficient conditions on a function \(f:(x,z)\in R^ n\times R^ n\mapsto f(x,z)\in [0,+\infty [\) measurable in the x variable and convex in the z one are also proposed in order to get identity, at least for every regular bounded open set \(\Omega\), u in \(W^{1,1}(\Omega)\), between the functionals \(F'(\Omega,u)=\int_{\Omega}f(x,Du)\) and \(F''(\Omega,u)=\inf \{\liminf_{h}\int_{\Omega}f(x,Du_ h)\), \(u_ h\) Lipschitz continuous, \(u_ h\to u\) in \(L^ 1(\Omega)\}\). In particular, if \(n=1\), identity between \(F'(\Omega,u)\) and \(F''(\Omega,u)\) for every bounded interval \(\Omega\), u in \(W^{1,1}(\Omega)\) is proved under very mild assumptions.
Reviewer: R.De Arcangelis


49J10 Existence theories for free problems in two or more independent variables
49J45 Methods involving semicontinuity and convergence; relaxation