No abstract

Let T ⊂ [a, b] be a time scale with a, b ∈ T. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem −u = λu σ , with mixed boundary conditions αu(a) + βu (a) = 0 = γ u(ρ(b)) + δu (ρ(b)). It is known that there exists a sequence of simple eigenvalues {λ k } k ; we consider the spectral counting function N(λ, T) = #{k: λ k λ}, and we seek for its asymptotic expansion as a power of λ. Let d be the Minkowski (or box) dimension of T, which gives the order of growth of the number K(T, ε) of intervals of length ε needed to cover T, namely K(T, ε) ≈ ε d . We prove an upper bound of N(λ) which involves the Minkowski dimension, N(λ, T) Cλ d/2 , where C is a positive constant depending only on the Minkowski content of T (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases (d = 0, infinite Minkowski content), and we show a family of self similar fractal sets where N(λ, T) admits two-side estimates.

In this article we prove modular and norm Pólya-Szegö inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of theses spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.This inequality is crucial in the proof of the Rayleigh-Faber-Krahn inequality, which asserts that balls minimize the first eigenvalue of the Dirichlet p−Laplacian among sets with given volume, that is,where B is a ball having the same measure as Ω. We refer the reader to the survey [24] for more information on the symmetric rearrangement.

Let T ¼ {a n } n < {0} be a time scale with zero Minkowski (or box) dimension, where {a n } n is a monotonically decreasing sequence converging to zero, and a 1 ¼ 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem 2 u DD ¼ lu s , with Dirichlet boundary conditions. We obtain that the ntheigenvalue is bounded below by 4a 22 n21 . We show that the bound is optimal for the qdifference equations arising in quantum calculus.