On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains. (English) Zbl 1350.49010
Summary: In this article, we give sufficient conditions for the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in a sequence of domains \(\Omega_s\) contained in a bounded domain \(\Omega\) of \(\mathbb{R}^n\) (\(n \geqslant 2\)). We study the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative measurable function on \(\Omega\). The statements of our main results include the condition of the \(\Gamma\)-convergence of the functionals (defined on the spaces \(W^{1, p}(\Omega_s)\)) to a functional defined on \(W^{1, p}(\Omega)\) and the condition of the strong connectedness of the spaces \(W^{1, p}(\Omega_s)\) with the space \(W^{1, p}(\Omega)\), where \(p > 1\). At the same time, because of the specificity of the imposed constraints, the exhaustion condition of the domain \(\Omega\) by the domains \(\Omega_s\) and the proposed requirement on the behavior of the integrands of the principal components of the considered functionals are also important for our convergence results.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
Keywords:
integral functional; bilateral problem; minimizers; \(\Gamma\)-convergence; strong connectednessReferences:
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