On the convergence of minimizers and minimum values in variational problems with pointwise functional constraints in variable domains

A. A. Kovalevsky

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 246-257

Résumé

We consider a sequence of convex integral functionals $F_s:W^{1,p}(\Omega_s)\to\mathbb R$ and a sequence of weakly lower semicontinuous and, in general, non-integral functionals $G_s:W^{1,p}(\Omega_s)\to\mathbb R$, where $\{\Omega_s\}$ is a sequence of domains in $\mathbb R^n$ contained in a bounded domain $\Omega\subset\mathbb R^n$ ($n\geqslant 2$) and $p>1$. Along with this, we consider a sequence of closed convex sets $V_s=\{v\in W^{1,p}(\Omega_s): M_s(v)\leqslant 0\,\,\text{a.e. in}\,\,\Omega_s\}$, where $M_s$ is a mapping from $W^{1,p}(\Omega_s)$ to the set of all functions defined on $\Omega_s$. We describe conditions under which minimizers and minimum values of the functionals $F_s+G_s$ on the sets $V_s$ converge to a minimizer and the minimum value of a functional on the set $V=\{v\in W^{1,p}(\Omega): M(v)\leqslant 0\,\,\text{a.e. in}\,\,\Omega\}$, where $M$ is a mapping from $W^{1,p}(\Omega)$ to the set of all functions defined on $\Omega$. In particular, for our convergence results, we require that the sequence of spaces $W^{1,p}(\Omega_s)$ is strongly connected with the space $W^{1,p}(\Omega)$ and the sequence $\{F_s\}$ $\it{\Gamma}$-converges to a functional defined on $W^{1,p}(\Omega)$. In so doing, we focus on the conditions on the mappings $M_s$ and $M$ which, along with the corresponding requirements on the involved domains and functionals, ensure the convergence of solutions of the considered variational problems. Such conditions have been obtained in our recent work, and, in the present paper, we advance in studying them.

Détail

Keywords: variational problem; integral functional; pointwise functional constraint; minimizer; minimum value; $\it{\Gamma}$-convergence; strong connectedness; variable domains.