In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional F defined on the family of all ’quasi-open’ subsets of a bounded open set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_315_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_315_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. This is ensured under the condition that F demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional p-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_315_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$s\rightarrow 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and extend this result to the anisotropic case.

Shape optimization problems involving nonlocal and nonlinear operators

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发表日期：

2025年2月15日

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Ignacio Ceresa Dussel

摘要

In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional F defined on the family of all ’quasi-open’ subsets of a bounded open set $\Omega $ $Ω$ in $\mathbb {R}^n$ $R^{n}$ . This is ensured under the condition that F demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional p-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $s\rightarrow 1$ $s \to 1$ and extend this result to the anisotropic case.