Convergence of solutions of variational problems with measurable bilateral constraints in variable domains

Abstract

We give conditions for the convergence of minimizers and minimum values of integral and more general functionals $J_s:W^{1,p}(\Omega_s)\rightarrow {\mathbb{R}}$ on sets of functions defined by a measurable lower constraint $\varphi :\Omega \rightarrow \overline{\mathbb{R}}$ and a measurable upper constraint $\psi :\Omega \rightarrow \overline{{\mathbb{R}}}$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ ($n\geqslant 2$), $\{\Omega_s\}$ is a sequence of domains in ${\mathbb{R}}^n$ contained in $\Omega$, and $p>1$. These conditions include the $\Gamma$-convergence of the functionals under consideration and some requirements on the domains $\Omega _s$. As for the constraints $\varphi$ and $\psi$, we consider two cases. In the first case, we assume that there exist functions $\bar{\varphi },\bar{\psi }\in W^{1,p}(\Omega )$ such that $\varphi \leqslant \bar{\varphi }<\bar{\psi }\leqslant \psi$ a.e. in $\Omega$. In the second case, we assume that there exists a function $\beta \in W^{1,p}(\Omega )$ satisfying the inequality $\varphi \leqslant \beta \leqslant \psi$ a.e. in $\Omega$ and interacting with the integrands of the main components of the functionals $J_s$ in a special way.