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Kovalevsky, Alexander A.

On the convergence of solutions of variational problems with pointwise functional constraints in variable domains. (English. Russian original) Zbl 1465.49012

J. Math. Sci., New York 254, No. 3, 375-396 (2021); translation from Ukr. Mat. Visn. 17, No. 4, 509-537 (2020).
Summary: We consider a sequence of convex integral functionals \(F_s : W^{1 ,p } ( \Omega_s) \rightarrow \mathbb{R}\) and a sequence of weakly lower semicontinuous and, in general, nonintegral functionals \(G_s : W^{1 ,p } ( \Omega_s) \rightarrow \mathbb{R} \), where \( \{\Omega_s\}\) is a sequence of domains in \(\mathbb{R}^n\) contained in a bounded domain \(\Omega \subset \mathbb{R}^n\) (\(n \geq 2\)) and \(p > 1\). Along with this, we consider a sequence of closed convex sets \(V_s = \{v \in W^{1 ,p } ( \Omega_s ) : M_s(v) \leq 0 \text{ a.e. in } \Omega_s \}\), where \(M_s\) is a mapping from \(W^{1 ,p } ( \Omega_s)\) to 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 of a functional on the set \(V = \{v \in W^{1 ,p } (\Omega ) : M(v) \leq 0 \text{ a.e. in } \Omega \}\), where \(M\) is a mapping from \(W^{1 ,p } (\Omega )\) to the set of all functions defined on \(\Omega \).

Cited in 1 Document

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation

Keywords:

variational problem; integral functional; pointwise functional constraint; minimizer; minimum value; \( \Gamma \)-convergence; strong connectedness; variable domains
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\textit{A. A. Kovalevsky}, J. Math. Sci., New York 254, No. 3, 375--396 (2021; Zbl 1465.49012); translation from Ukr. Mat. Visn. 17, No. 4, 509--537 (2020)
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References:

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