<p>In this paper we study Weyl-type eigenvalue bounds for the variational eigenvalues of the fractional <i>p</i>-Laplacian on a bounded domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2094_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\Omega \subset \mathbb{R}^{n}$</EquationSource> </InlineEquation> with a Lipschitz boundary. We prove that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2094_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>m</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>C</mi> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mi>s</mi> <mi>p</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </msup> <msup> <mi>m</mi> <mfrac> <mrow> <mi>s</mi> <mi>p</mi> </mrow> <mi>n</mi> </mfrac> </msup> </math></EquationSource> <EquationSource Format="TEX">$\lambda _{m}(\Omega )\geq C |\Omega |^{-\frac{sp}{n}} m^{\frac{sp}{n}}$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2094_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>C</mi> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$C=C(s,p,n)$</EquationSource> </InlineEquation>. This result partially confirms a conjecture of Iannizzotto and Squassina (Asymptot. Anal. 88: 233–245, <CitationRef CitationID="CR16">2014</CitationRef>). We also study the auxiliary weighted eigenvalue problem (<InternalRef RefID="Equ2">1.2</InternalRef>) and obtain that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2094_Article_IEq4.gif" Format="GIF" Height="39" Rendition="HTML" Resolution="72" Type="Linedraw" Width="296" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>m</mi> </msub> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>C</mi> <msup> <mrow> <mo maxsize="3.8ex" minsize="3.8ex" stretchy="true">(</mo> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <msup> <mi>w</mi> <mi>r</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo maxsize="3.8ex" minsize="3.8ex" stretchy="true">)</mo> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </msup> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mo>−</mo> <mfrac> <mrow> <mi>s</mi> <mi>p</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </msup> <msup> <mi>m</mi> <mfrac> <mrow> <mi>s</mi> <mi>p</mi> </mrow> <mi>n</mi> </mfrac> </msup> </math></EquationSource> <EquationSource Format="TEX">$\lambda _{m}(w, \Omega )\geq C\Bigl(\int _{\Omega }w^{r}(x)dx \Bigr)^{-\frac{1}{r}}| \Omega |^{\frac{1}{r}-\frac{sp}{n}} m^{\frac{sp}{n}}$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2094_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>C</mi> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$C=C(s,p,n,r)$</EquationSource> </InlineEquation>. The main method used in this paper is the approximation method developed by Birman and Solomyak (Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory, <CitationRef CitationID="CR1">1980</CitationRef>).</p>

Weyl-type eigenvalue bounds for the fractional p-Laplacian

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2025年7月10日

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Mahir Hasanov

摘要

In this paper we study Weyl-type eigenvalue bounds for the variational eigenvalues of the fractional p-Laplacian on a bounded domain $Ω \subset R^{n}$ $\Omega \subset \mathbb{R}^{n}$ with a Lipschitz boundary. We prove that $λ_{m} (Ω) \geq C | Ω |^{- \frac{s p}{n}} m^{\frac{s p}{n}}$ $\lambda _{m}(\Omega )\geq C |\Omega |^{-\frac{sp}{n}} m^{\frac{sp}{n}}$ , where $C = C (s, p, n)$ $C=C(s,p,n)$ . This result partially confirms a conjecture of Iannizzotto and Squassina (Asymptot. Anal. 88: 233–245, 2014). We also study the auxiliary weighted eigenvalue problem (1.2) and obtain that $λ_{m} (w, Ω) \geq C {(\int_{Ω} w^{r} (x) d x)}^{- \frac{1}{r}} | Ω |^{\frac{1}{r} - \frac{s p}{n}} m^{\frac{s p}{n}}$ $\lambda _{m}(w, \Omega )\geq C\Bigl(\int _{\Omega }w^{r}(x)dx \Bigr)^{-\frac{1}{r}}| \Omega |^{\frac{1}{r}-\frac{sp}{n}} m^{\frac{sp}{n}}$ , where $C = C (s, p, n, r)$ $C=C(s,p,n,r)$ . The main method used in this paper is the approximation method developed by Birman and Solomyak (Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory, 1980).