Variational problems with unilateral pointwise functional constraints in variable domains

We consider a sequence of convex integral functionals $F_s\colon W^{1,p}(\Omega_s)\to\mathbb R$ and a sequence of weakly lower semicontinuous and, in general, non-integral functionals $G_s\colon W^{1,p}(\Omega_s)\to\mathbb R$, where $\{\Omega_s\}$ is a sequence of domains of $\mathbb R^n$ 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 $V_s=\{v\in W^{1,p}(\Omega_s)\colon v\geqslant K_s(v)\text{ a.e. in }\Omega_s\}$, where $K_s$ is a mapping of the space $W^{1,p}(\Omega_s)$ into 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, respectively, of a certain functional on the set $V=\{v\in W^{1,p}(\Omega)\colon v\geqslant K(v)\text{ a.e. in }\Omega\}$, where $K$ is a mapping of the space $W^{1,p}(\Omega)$ into the set of all functions defined on $\Omega$. These conditions include, in particular, the strong connectedness of the spaces $W^{1,p}(\Omega_s)$ with the space $W^{1,p}(\Omega)$, the exhaustion condition of the domain $\Omega$ by the domains $\Omega_s$, the $\Gamma$-convergence of the sequence $\{F_s\}$ to a functional $F\colon W^{1,p}(\Omega)\to\mathbb R$, and a certain convergence of the sequence $\{G_s\}$ to a functional $G\colon W^{1,p}(\Omega)\to\mathbb R$. We also assume certain conditions that characterize both the internal properties of the mappings $K_s$ and their relation to the mapping $K$. In particular, these conditions admit the study of variational problems with unilateral varying irregular obstacles and with varying constraints combining the pointwise dependence and the functional dependence of the integral form.