## 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

### MSC:

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

### Keywords:

relaxation; lower semicontinuity; Lavrentieff phenomenon