Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

Abstract

Abstract. We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells.These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).

Spectral geometry in a rotating frame: Properties of the ground state

“…It is obvious that without loss of generality we may put x 0 = y 0 = 0. The quadratic form corresponding to H ω,Ω (0, 0) is (20) Q(H ω,Ω (0, 0))(u) = We are going to pass to new coordinates analogous to those used in [2]) changing the radial one to r = tR(ϕ) with t ∈ [0, 1]. This allows to rewrite the form (20) as…”

“…which is also the natural domain of the operator (1). Furthermore, the vector potential 1 2 ω gives rise to a homogeneous 'magnetic field', and as a consequence,Ĥ ω (x 0 , y 0 ) is a by simple gauge transformation, u(x, y) → u(x, y)e −iω(xy 0 −yx 0 )/2 , unitarily equivalent to (2) H…”

“…with A := (−y, x); in what follows we will work with both the operators (1) and (2). The boundedness of the second term at the right-hand side together with the 'magnetic' form of the first also implies that the spectrum of H ω (x 0 , y 0 ) is purely discrete and the eigenvalues obey for any fixed ω > 0 and (x 0 , y 0 ) ∈ R 2 Weyl's law,…”

The Robin Laplacian—Spectral conjectures, rectangular theorems

“…Although Bareket's conjecture turns out to be false for large α < 0 due to an annular counterexample by Freitas and Krejčiřík [19], those authors did establish the conjecture in 2 dimensions whenever α < 0 is small enough, depending only on the volume of the domain. Bareket's conjecture holds also when the domain is close enough to a ball, by Ferone, Nitsch and Trombetti [17], and holds for all domains in 2-dimensions if perimeter rather than area of the domain is normalized, by Antunes, Freitas and Krejčiřík [4,Theorem 2].…”

A Sharp estimate for the first Robin–Laplacian eigenvalue with negative boundary parameter

“…In 2017, Antunes, Freitas and Krejcirik studied (see [2]) the problem of maximising the first eigenvalue under a perimeter constraint, proving that the disc is the solution among all C 2 domains of R 2 . Their proof is based on a comparison argument obtained by the method of parallel coordinates, originally introduced by Payne and Weinberger [16].…”