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Isoperimetric Sets for Weighted Twisted Eigenvalues

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Abstract

In this paper we prove an isoperimetric inequality for the first twisted eigenvalue \(\lambda _{1,\gamma }^T(\Omega )\) of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space \(H_0^1(\Omega ,{d}\gamma )\) and have weighted mean value equal to zero in \(\Omega \). We are interested in positive measures \({d}\gamma =\gamma (x) {d}x\) for which we are able to identify the optimal sets, namely, the sets that minimize \(\lambda _{1,\gamma }^T(\Omega )\) among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of the isoperimetric sets (for the weighted perimeter with respect to the weighted measure).

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Acknowledgements

The authors want to warmly thank the three referees who allow them to improve the quality of the paper. A.H. wants to thank the Dipartimento di Matematica e Applicazioni “R. Caccioppoli” for a nice and fruitful stay in July 2022. B.B. has been supported by “FFR2023 Barbara Brandolini”, Università degli Studi di Palermo. A.M. and M.R.P. have been partially supported by MUR through research project PRIN2017 “Direct and Inverse problems for PDE: theoretical aspects and applications”.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, via Archirafi 34, 90123, Palermo, Italy

    Barbara Brandolini

  2. CNRS, Institut Elie Cartan de Lorraine, Université de Lorraine, 54000, Nancy, France

    Antoine Henrot

  3. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Monte S. Angelo - via Cintia, 80126, Napoli, Italy

    Anna Mercaldo & Maria Rosaria Posteraro

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  1. Barbara Brandolini
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  2. Antoine Henrot
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  4. Maria Rosaria Posteraro
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Correspondence to Antoine Henrot.

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Brandolini, B., Henrot, A., Mercaldo, A. et al. Isoperimetric Sets for Weighted Twisted Eigenvalues. J Geom Anal 33, 361 (2023). https://doi.org/10.1007/s12220-023-01420-9

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