Principal frequency of Δ_∞ as limit of Rayleigh quotients in Orlicz spaces

Abstract

The asymptotic behavior of Rayleigh quotients involving Luxemburg norms in Orlicz spaces $L^{{\mathrm{\Psi }}_n}$ is studied as $n\to \mathrm{\infty }$, when $({\mathrm{\Psi }}_n{)}_n$ is a sequence of N-functions. In a particular case, we recover a well-known result of Juutinen, Lindqvist & Manfredi and Fukagai, Ito & Narukawa regarding the limit, as $p\to \mathrm{\infty },$ of the minima of Rayleigh quotients associated to the eigenvalue problem for the p-Laplacian.

Keywords:

• Luxemburg norm
• principal frequency
• Rayleigh quotient
• Orlicz and Orlicz–Sobolev spaces

2010 Mathematics Subject Classifications:

• 35P30
• 35D40
• 46E30
• 49J40
• 49J45

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Mihai Mihăilescu was partially supported by Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii (CNCS-UEFISCDI) [grant number PN-III-P4-ID-PCE-2016-0035].