This paper considers the Dirichlet eigenvalue problem of one-dimensional p–Laplacian with an integrable potential <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$q\in {\mathcal {L}}^1(I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. By viewing eigenvalues as nonlinear functionals of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q\in {\mathcal {L}}^1(I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we will mainly study the maximization problem on the gaps between even–order eigenvalues and the first eigenvalue when the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {L}}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norm of q is bounded by some given constant. In order to characterize the maximizer, we first extend the domain of these functionals from integrable potentials to singular potential measures. Then by employing the solutions to some optimization problems studied in previous papers, we will completely solve the maximization problem.

Maximal Gaps of Eigenvalues of One-Dimensional p-Laplacian

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2026年1月16日

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Zhiyuan Wen,
Lijuan Zhou

摘要

This paper considers the Dirichlet eigenvalue problem of one-dimensional p–Laplacian with an integrable potential $q\in {\mathcal {L}}^1(I)$ $q \in L^{1} (I)$ . By viewing eigenvalues as nonlinear functionals of $q\in {\mathcal {L}}^1(I)$ $q \in L^{1} (I)$ , we will mainly study the maximization problem on the gaps between even–order eigenvalues and the first eigenvalue when the ${\mathcal {L}}^1$ $L^{1}$ norm of q is bounded by some given constant. In order to characterize the maximizer, we first extend the domain of these functionals from integrable potentials to singular potential measures. Then by employing the solutions to some optimization problems studied in previous papers, we will completely solve the maximization problem.