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

We prove explicit and sharp eigenvalue estimates for Neumann p-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if gamma denotes a non-closed curve in R-2 symmetric with respect to the y-axis, let D subset of R-2 denote the domain of points that lie on one side of gamma and within a prescribed distance delta(s) from gamma (s) (here s denotes the arc length parameter for gamma). Write mu odd 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 mu (odd)(1) (D) when the distance function delta and the signed curvature k of gamma satisfy certain geometric constraints. In the linear case (p = 2), we establish sufficient conditions to guarantee mu (odd)(1) (D) = mu (1)(D). We finally study the asymptotics of mu (1)(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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data