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On spectral optimization for p-Laplace type operators

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Raul Fernandes Horta, Marcos Montenegro

Analysis of PDEs · Submitted 2 May 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Sharp lower and upper uniform estimates are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms.

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arXiv ID2405.01457 (version 3)
Pith IDUIV33AZA, Pith's permanent record of this paper and its review
CitationMarcos Montenegro, Raul Fernandes Horta. "On spectral optimization for p-Laplace type operators." arXiv:2405.01457 (2024). https://pith.science/paper/UIV33AZA
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,297 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.57 · compares with prior results 0.59 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

Sharp lower and upper uniform estimates are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. Optimal constants are exhibited, rigidity of the upper estimate is proved, anisotropic attainability of the lower estimate is derived as well as characterization of anisotropic extremizers for circular and rectangular membranes. Sharp quantitative anisotropic inequalities associated with lower constants are also established, providing as a by-product information on anisotropic stability. When the uniform ellipticity condition is relaxed, we show that the optimal lower constant remains positive, while anisotropic extremizers no longer exist. Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the Faber-Krahn isoperimetric inequality in the plane.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Sharp lower and upper uniform estimates are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. result 0.94 0.93 0.64 asserted 1.00
s1 Optimal constants are exhibited, rigidity of the upper estimate is proved, anisotropic attainability of the lower estimate is derived as well as characterization of anisotropic extremizers for circular and rectangular membranes. result 0.75 0.95 0.34 asserted 1.00
s2 Sharp quantitative anisotropic inequalities associated with lower constants are also established, providing as a by-product information on anisotropic stability. result 0.93 0.93 0.14 asserted 1.00
s3 When the uniform ellipticity condition is relaxed, we show that the optimal lower constant remains positive, while anisotropic extremizers no longer exist. result 0.96 0.95 0.12 asserted 1.00
s4 Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the Faber-Krahn isoperimetric inequality in the plane. interpretation 1.00 0.70 0.23 asserted 0.67

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Original abstract

Sharp uniform estimates, both lower and upper, are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. Optimal constants are computed, rigidity of the upper estimate is derived, anisotropic attainability of the lower constant is established, along with the characterization of corresponding anisotropic extremizers for certain membranes. Sharp quantitative inequalities associated with lower constants are also proved, thereby providing anisotropic stability. The optimal lower constant remains positive even when the condition of uniform strict ellipticity of the operators is removed. Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the two-dimensional Faber-Krahn isoperimetric inequality.

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