Basis properties of eigenfunctions the one-dimensional p-Laplacian

7 March 2007, 3 pm

 

Speaker: Dr L Boulton (Heriot-Watt)

Generalised sine functions are defined as the Dirichlet eigenfunctions associated to the first eigenvalue of the one-dimensional $p$-Laplacian equation on an interval of length π_p := 2 π/(p sin(π/p)). In the early eighties various properties of these functions were studied. Among others, p ≠ 2 versions of the Pythagorean relation and characterisations of the eigenfunctions associated to higher order eigenvalues. Despite this activity, it seems that analogues for p ≠ 2 of the standard completeness and expansion theorems for sine functions have not been considered previously. In this talk we will show that for 12/11 ≤ p < ∞, the family of eigenfunctions of the p-Laplacian forms a Schauder basis of L^2(0, π_p).