Eigenvalue estimates for p-Laplace problems on domains expressed in Fermi coordinates

Abstract

We prove explicit and sharp eigenvalue estimates for Neumann p-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if γ denotes a non-closed curve in R2 symmetric with respect to the y-axis, let D⊂R2 denote the domain of points that lie on one side of γ and within a prescribed distance δ(s) from γ(s) (here s denotes the arc length parameter for γ). Write μ1odd(D) for the lowest nonzero eigenvalue of the Neumann p-Laplacian with an eigenfunction that is odd with respect to the y-axis. For all p>1, we provide a lower bound on μ1odd(D) when the distance function δ and the signed curvature k of γ satisfy certain geometric constraints. In the linear case (p=2), we establish sufficient conditions to guarantee μ1odd(D)=μ1(D). We finally study the asymptotics of μ1(D) as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann p-Laplace problem.