Abstract:
We consider a sequence of convex integral functionals Fs:W1,p(Ωs)→R and a sequence of weakly lower semicontinuous and, in general, non-integral functionals Gs:W1,p(Ωs)→R, where {Ωs} is a sequence of domains in Rn contained in a bounded domain Ω⊂Rn (n⩾2) and p>1. Along with this, we consider a sequence of closed convex sets Vs={v∈W1,p(Ωs):Ms(v)⩽0a.e. inΩs}, where Ms is a mapping from W1,p(Ωs) to the set of all functions defined on Ωs. We describe conditions under which minimizers and minimum values of the functionals Fs+Gs on the sets Vs converge to a minimizer and the minimum value of a functional on the set V={v∈W1,p(Ω):M(v)⩽0a.e. inΩ}, where M is a mapping from W1,p(Ω) to the set of all functions defined on Ω. In particular, for our convergence results, we require that the sequence of spaces W1,p(Ωs) is strongly connected with the space W1,p(Ω) and the sequence {Fs}Γ-converges to a functional defined on W1,p(Ω). In so doing, we focus on the conditions on the mappings Ms and M which, along with the corresponding requirements on the involved domains and functionals, ensure the convergence of solutions of the considered variational problems. Such conditions have been obtained in our recent work, and, in the present paper, we advance in studying them.
This work was partially supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).
Citation:
A. A. Kovalevsky, “On the convergence of minimizers and minimum values in variational problems with pointwise functional constraints in variable domains”, Trudy Inst. Mat. i Mekh. UrO RAN, 27, no. 1, 2021, 246–257
\Bibitem{Kov21}
\by A.~A.~Kovalevsky
\paper On the convergence of minimizers and minimum values in variational problems with pointwise functional constraints in variable domains
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2021
\vol 27
\issue 1
\pages 246--257
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This publication is cited in the following 1 articles:
Alexander A. Kovalevsky, “Approximation in W1,p-norms of solutions of minimum problems with bilateral constraints in variable domains”, Boll Unione Mat Ital, 2025