Convergence of solutions of bilateral problems in variable domains and related questions

Alexander A. Kovalevsky

Ural mathematical journal, Tome 3 (2017) no. 2, pp. 51-66

Résumé

We discuss some results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. The first case concerns a larger class of integrands and requires the positivity almost everywhere of the difference between the upper and lower constraints. In the second case, this requirement is absent. Moreover, in the latter case, the exhaustion condition of an $n$-dimensional domain by a sequence of $n$-dimensional domains plays an important role. We give a series of results involving this condition. In particular, using the exhaustion condition, we prove a certain convergence of sets of functions defined by bilateral (generally irregular) constraints in variable domains.

Détail

Keywords: Integral functional, Bilateral problem, Minimizer, Minimum value, $\Gamma$-convergence of functionals, Strong connectedness of spaces, $\mathcal H$-convergence of sets, Exhaustion condition.