Monotonicity properties for the variational Dirichlet eigenvalues of the p-Laplace operator

Abstract

Let D≥2 be an integer. For each open and bounded set Ω⊂RD and each integer k≥1 we denote by Rk(Ω) the largest number r>0 for which there exists k disjoint open balls in Ω of radius r. Next, for each open, bounded, convex set Ω⊂RD with smooth boundary and each real number p∈(1,∞) we denote by {λk(p;Ω)}k≥1 the sequence of eigenvalues of the p-Laplace operator subject to the homogeneous Dirichlet boundary conditions, given by the Ljusternik-Schnirelman theory. For each integer k≥1 we show that there exists Mk∈[(ke)−1,1] such that for any open, bounded, convex set Ω⊂RD with smooth boundary for which Rk(Ω) is less than or equal to Mk, the k-th eigenvalue of the p-Laplacian on Ω, λk(p;Ω), is an increasing function of p on (1,∞). Moreover, there exists Nk≥Mk such that for any real number s∈(Nk,∞)∖{1} there exists an open, bounded, convex set Ω⊂RD with smooth boundary which has Rk(Ω) equal to s such that λk(p;Ω) is not a monotone function of p∈(1,∞).