Isolation and simplicity for the first eigenvalue of the p‐Laplacian with a nonlinear boundary condition

Abstract

We prove the simplicity and isolation of the first eigenvalue for the problem ∆ p u = |u| p−2 u in a bounded smooth domain Ω ⊂ R N , with a nonlinear boundary condition given by |∇u| p−2 ∂u/∂ν = λ|u| p−2 u on the boundary of the domain.

Asymptotic Behavior of the Steklov Eigenvalues For the p−Laplace Operator

“…Also, it is not difficult to show that for both problems the first eigenvalue is positive and the associate eigenfunction is nonnegative. We leave the details for the reader, although they can be grasped from [7] and [15].…”

“…However, few facts are known about this sequence. It is not known if this variational sequence exhausts the spectrum, although it was proved that λ 1 is the only eigenvalue with a positive eigenfunction, it is simple, and there are no other eigenvalues between λ 1 and λ 2 , see [15]. Moreover, it can be obtained by minimization of the Rayleigh quotient over all of W 1,p (Ω):…”

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

“…Then we consider A = B r . Since the first eigenvalue λ 1 (A) is simple (see [18]), it follows that u is radial. Hence there exists a constant C such that at the boundary of the hole we have, ∂u ∂ν = C. Therefore, using that we are dealing with deformations V that preserves the volume of the hole, we get…”

“…It at least goes back to [1], for more references see [5]. In particular, the Sobolev trace inequality has been intensively studied in [2,6,7,8,9,10,16,17,18], etc.…”

Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains

“…For p = q it is proved in [12] that the first eigenvalue λ 1 (Ω µ ) = S p (Ω µ ) is isolated and simple. As a consequence of this if Ω is a ball the extremal v µ is radial and hence it does not develop a peak.…”