Variational problems with unilateral pointwise functional constraints in variable domains

A. A. Kovalevsky

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 133-150

Résumé

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.

Détail

Mots-clés : variable domains, $\Gamma$-convergence
Keywords: integral functional, unilateral pointwise functional constraints, minimizer, minimum value, strong connectedness.

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