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<p>We consider a sequence of minimum problems for integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. Under some conditions on the involved domains, functionals, and constraints, we prove that the sequence of minimizers of the considered problems is approximated in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_467_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(W^{1,p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-norms by a special <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_467_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varGamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Γ</mi> </math></EquationSource> </InlineEquation>-realizing sequence for the minimizer of the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_467_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varGamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Γ</mi> </math></EquationSource> </InlineEquation>-limit functional on a limit set. This <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_467_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varGamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Γ</mi> </math></EquationSource> </InlineEquation>-realizing sequence depends on the given constraints and each its element belongs to the corresponding constraint set. The crucial role in obtaining our approximation result is played by the assumption that the considered sequence of functionals satisfies the uniform convexity condition.</p>

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Approximation in \(W^{1,p}\)-norms of solutions of minimum problems with bilateral constraints in variable domains

  • 期刊论文

  • 发表日期：

    2025年2月15日

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  • Alexander A. Kovalevsky

摘要

We consider a sequence of minimum problems for integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. Under some conditions on the involved domains, functionals, and constraints, we prove that the sequence of minimizers of the considered problems is approximated in \(W^{1,p}\) W 1 , p -norms by a special \(\varGamma \) Γ -realizing sequence for the minimizer of the \(\varGamma \) Γ -limit functional on a limit set. This \(\varGamma \) Γ -realizing sequence depends on the given constraints and each its element belongs to the corresponding constraint set. The crucial role in obtaining our approximation result is played by the assumption that the considered sequence of functionals satisfies the uniform convexity condition.