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On the first eigenvalue of the normalized $ p$-Laplacian


Authors: Graziano Crasta, Ilaria Fragalà and Bernd Kawohl
Journal: Proc. Amer. Math. Soc. 148 (2020), 577-590
MSC (2010): Primary 49K20, 35J60, 47J10
DOI: https://doi.org/10.1090/proc/14823
Published electronically: November 6, 2019
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Abstract: We prove that if $ \Omega $ is an open bounded domain with smooth and connected boundary, for every $ p \in (1, + \infty )$ the first Dirichlet eigenvalue of the normalized $ p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (nonoptimal) lower bound for the eigenvalue in terms of the measure of $ \Omega $, and we address the open problem of proving a Faber-Krahn-type inequality with balls as optimal domains.


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Additional Information

Graziano Crasta
Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I Piazzale Aldo Moro 2 – 00185 Roma, Italy
Email: crasta@mat.uniroma1.it

Ilaria Fragalà
Affiliation: Dipartimento di Matematica, Politecnico Piazza Leonardo da Vinci, 32 –20133 Milano, Italy
Email: ilaria.fragala@polimi.it

Bernd Kawohl
Affiliation: Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany
Email: kawohl@math.uni-koeln.de

DOI: https://doi.org/10.1090/proc/14823
Keywords: Normalized $p$-Laplacian, viscosity solutions, eigenvalue problem.
Received by editor(s): January 16, 2019
Published electronically: November 6, 2019
Additional Notes: The first and second authors were supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
Communicated by: Joachim Krieger
Article copyright: © Copyright 2019 American Mathematical Society