Basis properties of eigenfunctions of the 𝑝-Laplacian

Abstract

Abstract. For p 12 11 , the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0, 1) are shown to form a Riesz basis of L 2 (0, 1) and a Schauder basis of L q (0, 1) whenever 1 < q < ∞.

Decay of (p,q)-Fourier coefficients

“…The b k,p,q may be thought of as p, q-Fourier coefficients of f : note that when p = 2, the lack of orthogonality of the f k,p,q means that the b k,p,q do not have the attractive integral representation possessed by the e k . Proofs of these basis assertions rely on a technique originally adopted in [4] and which we now outline. Given any f :…”

On asymptotic behavior of Dirichlet inverse

“…3.3.2 f (n) supported on odd n. Assume that f (n) represents the Fourier coefficients of a function F ∈ L 2 (0, 1), that is, F (x) = ∞ n=1 f (n) sin(nπx). If one is interested in the basisness or completeness of the system {F (nx)} (see Section 1), then it seems natural to choose F such that it is symmetric with respect to the point x = 1/2, see, e.g., generalized trigonometric functions [2]. Under this symmetry assumption, we have f (n) = 0 for all even n and it is clear from (1.3) that f −1 (n) = 0 for all even n, as well.…”

Basis properties of the p , q -sine functions

“…In fact, more rudimentary methods can be invoked in order to examine the invertibility of the change of coordinates map. From ( 1.1 ), it follows that In Binding et al [ 5 ], it was claimed that the left-hand side of ( 1.2 ) held true for all p = q ≥ p 1 , where p 1 was determined to lie in the segment . Hence, would be a Schauder basis, whenever .…”

“…Among these properties lie the fundamental question of completeness and linear independence of the family , where . This question has received some attention recently [ 2 – 5 ], with a particular emphasis on the case p = q . In the latter instance, is the set of eigenfunctions of the generalized eigenvalue problem for the one-dimensional p -Laplacian subjected to Dirichlet boundary conditions [ 6 , 7 ], which is known to be of relevance in the theory of slow/fast diffusion processes, [ 8 ].…”

“…Further developments in this respect were recently reported by Bushell & Edmunds [ 4 ]. These authors cleverly fixed a gap originally published in [ 5 ], lemma 5 and observed that, as the left-hand side of ( 1.2 ) ceases to hold true whenever the argument will break for p = q near p 2 ≈1.043989. Therefore, the basisness question for should be tackled by different means in the regime p , q →1.…”