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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli
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by T. V. Anoop, Vladimir Bobkov and Sarath Sasi PDF
Trans. Amer. Math. Soc. 370 (2018), 7181-7199 Request permission


Let $B_1$ be a ball in $\mathbb {R}^N$ centred at the origin and let $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in (1, \infty )$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline {B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p \to 1$ and $p \to \infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the $p$-Laplacian on bounded radial domains.
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Additional Information
  • T. V. Anoop
  • Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
  • Email:
  • Vladimir Bobkov
  • Affiliation: Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeň 306 14, Czech Republic — and — Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa 450008, Russia
  • MR Author ID: 1040393
  • Email:
  • Sarath Sasi
  • Affiliation: School of Mathematical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni 752050, India
  • Address at time of publication: Indian Institute of Technology Palakkad, Ahalia Integrated Campus, Kozhipara, Palakkad 678557, Kerala, India
  • Email:
  • Received by editor(s): November 10, 2016
  • Received by editor(s) in revised form: March 15, 2017
  • Published electronically: June 26, 2018
  • Additional Notes: The second author was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 7181-7199
  • MSC (2010): Primary 35J92, 35P30, 35B06, 49R05
  • DOI:
  • MathSciNet review: 3841846