On the convergence of solutions of variational problems with pointwise functional constraints in variable domains

Abstract

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, nonintegral 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 establish 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 $\Omega$.