On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains

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• 作者：Alexander A. Kovalevsky ; ^{alexkvl77@mail.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：49J40 ; 49J45
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：147
• 期：Complete
• 页码：63-79
• 全文大小：681 K

文摘

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 Ω_s${\Omega }_s$ contained in a bounded domain 03a036e20a5b409e332bf" title="Click to view the MathML source">Ω$\Omega$ of Rⁿ${\mathbb{R}}^n$ (n⩾2$n⩾2$). We study the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative measurable function on 03a036e20a5b409e332bf" title="Click to view the MathML source">Ω$\Omega$. The statements of our main results include the condition of the a05e88219beee5ad09739b8c0c8f0" title="Click to view the MathML source">Γ$\Gamma$-convergence of the functionals (defined on the spaces W^1,p(Ω_s)$W^{1,p}\left({\Omega }_s\right)$) to a functional defined on W^1,p(Ω)$W^{1,p}\left(\Omega \right)$ and the condition of the strong connectedness of the spaces W^1,p(Ω_s)$W^{1,p}\left({\Omega }_s\right)$ with the space W^1,p(Ω)$W^{1,p}\left(\Omega \right)$, where p>1$p>1$. At the same time, because of the specificity of the imposed constraints, the exhaustion condition of the domain 03a036e20a5b409e332bf" title="Click to view the MathML source">Ω$\Omega$ by the domains Ω_s${\Omega }_s$ 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.