The principal frequency of Δ∞ as a limit of Rayleigh quotients involving Luxemburg norms

The asymptotic behavior of Rayleigh quotients involving both Luxemburg norms and modulars in the variable exponent Lebesgue space Lp(⋅) is studied as p(⋅)→∞. In a particular case, we recover a well-known result of Juutinen, Lindqvist and Manfredi regarding the limit, as p→∞ of the minima of Rayleigh quotients associated to the eigenvalue problem for the p-Laplacian.