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Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures

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Marco Degiovanni, Dario Mazzoleni

Analysis of PDEs · Submitted 23 May 2019

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

The authors' main claim In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure.

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arXiv ID1905.09563 (version 2)
Pith IDSFJKB4H2, Pith's permanent record of this paper and its review
CitationDario Mazzoleni, Marco Degiovanni. "Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures." arXiv:1905.09563 (2019). https://pith.science/paper/SFJKB4H2
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,104 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.08 · mentions code or data availability 0.02.

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

In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. result 0.83 0.96 0.79 asserted 1.00
s1 An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. result 0.93 0.91 0.09 asserted 1.00
s2 In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence. method 0.94 0.95 0.12 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.

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