On the variational convergence of non-coercive quadratic integral functionals and semicontinuity problems
The paper is concerned with the relationship between pointwise convergence of a sequence of function matrices, and the \(\Gamma\)-convergence of the corresponding quadratic functionals. Let \(\Omega\) be a bounded open subset of \({\mathbb R}^n\), let \(\{a_{ij}^h\}_{i,j\in\{1,\dots,n\}}\), \(h\in{\mathbb N}\), \(\{a_{ij}\}_{i,j\in\{1,\dots,n\}}\) be function matrices defined in \(\Omega\). First of all, it is recalled that the uniform convergence of \(a_{ij}^h\) to \(a_{ij}\) implies the sequential \(\Gamma\)-convergence in the weak-\(L^1(\Omega)\) topology of the functionals \[ F_h\colon u\in W^{1,2}(\Omega)\mapsto\int_\Omega\sum_{i,j=1}^n a_{ij}^h(x)D_iu(x)D_ju(x)dx \] to \[ F: u\in W^{1,2}(\Omega)\mapsto\int_\Omega\sum_{i,j=1}^n a_{ij}(x)D_iu(x)D_ju(x)dx, \] provided the \(\{a_{ij}^h\}\) are bounded, and are equicoercive, i.e., satisfy \(0<\lambda|z|^2\leq\sum_{i,j=1}^n a_{ij}^h(x)z_iz_j\) for every \(z\in{\mathbb R}^n\) and some \(\lambda>0\). Then, an example is produced showing that the above statement can be false if the equicoerciveness condition is dropped, even if it is replaced by \[ 0\leq\sum_{i,j=1}^n a_{ij}^h(x)z_iz_j\text{ for a.e. }x\in\Omega\text{ and every }z\in{\mathbb R}^n. \] Under this last condition, a sufficient condition to guarantee the sequential \(\Gamma\)-convergence of the functionals \(F_h\) to \(F\) in the weak-\(L^1(\Omega)\) topology is proved. It is shown that this holds provided the \(a_{ij}^h\)s are equibounded, converge a.e. to \(a_{ij}\), and the divergences of the column vectors of \(\{a_{i_j}^h\}\) are locally equibounded and converge a.e. to the divergences of the corresponding column vectors of \(\{a_{ij}\}\). The case of \(\Gamma\)-convergence in the strong-\(L^1(\Omega)\) topology is also treated. It is proved that, if \(a_{ij}^h\), \(a_{ij}\in L^\infty(\Omega)\), \(a_{ij}^h\) converge to \(a_{ij}\) in weak-\(L^1(\Omega)\), and if the divergences of the column vectors of \(\{a_{ij}^h\}\) converge to the divergences of the corresponding column vectors of \(\{a_{ij}\}\) weakly in \(L^1_{\text{ loc}}(\Omega)\), then the \(F_h\)s \(\Gamma\)-converge to \(F\) in the strong-\(L^1(\Omega)\) topology. An example proves that it is not possible to replace the strong-\(L^1(\Omega)\) topology with the weak one in the above statement. Finally, the case of sequential \(\Gamma\)-convergence in the weak-\(W^{1,2}(\Omega)\) topology is also treated. It is proved that, if \(a_{ij}^h\), \(a_{ij}\in L^\infty(\Omega)\), \(a_{ij}^h\) converge to \(a_{ij}\) in weak*-\(L^\infty(\Omega)\), and if the divergences of the column vectors of \(\{a_{ij}^h\}\) converge to the divergences of the corresponding column vectors of \(\{a_{ij}\}\) strongly in \(W^{-1,2}_{\text{ loc}}(\Omega)\), then the \(F_h\)s \(\Gamma\)-converge to \(F\) in the strong-\(L^1(\Omega)\) topology. Also in this case, an example proves that it is not possible to replace the weak-\(W^{1,2}(\Omega)\) topology with the strong-\(L^1(\Omega)\) one in the above statement. An \(L^1(\Omega)\)-lower semicontinuity result is also deduced as corollary. Eventually, some \(\Gamma\)-convergence results in the weak-\(L^1(\Omega)\) topology are established under a uniform convergence of the \(a_{ij}^h\)s to \(a_ij\) assumption, provided \[ 0\leq a(x)|z|^2\leq\sum_{i,j=1}^n a_{ij}^h(x)z_iz_j\leq k(x)a(x)\text{ for a.e. }x\in\Omega\text{ and every }z\in{\mathbb R}^n, \] where \(a\) is lower semicontinuous. An example shows that if \(a\) is upper semicontinuous, then the above \(\Gamma\)-convergence result does not hold, even if strong-\(L^1(\Omega)\) convergence is taken into account.
scientific article; zbMATH DE number 1982559