Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

On the strict monotonicity of the first eigenvalue of the $ p$-Laplacian on annuli


Authors: T. V. Anoop, Vladimir Bobkov and Sarath Sasi
Journal: Trans. Amer. Math. Soc. 370 (2018), 7181-7199
MSC (2010): Primary 35J92, 35P30, 35B06, 49R05
DOI: https://doi.org/10.1090/tran/7241
Published electronically: June 26, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35J92, 35P30, 35B06, 49R05

Retrieve articles in all journals with MSC (2010): 35J92, 35P30, 35B06, 49R05


Additional Information

T. V. Anoop
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Email: anoop@iitm.ac.in

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
Email: bobkov@kma.zcu.cz

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: sarath@iitpkd.ac.in

DOI: https://doi.org/10.1090/tran/7241
Keywords: $p$-Laplacian, symmetries, shape derivative, Fu\v{c}ik spectrum, eigenvalue, eigenfunction, nonradiality
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.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society