An extremal eigenvalue problem for the Wentzell–Laplace operator

Abstract

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain Ω, involving only geometrical information. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

“…To compute the derivatives of σ andσ with respect to the centers c and radii r, we first compute the shape derivative with respect to perturbations of the boundary of Ω c,r . This result extends a result in [1,6,9] to ρ = 1.…”

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

“…We denote by ∆ and ∆ the Laplace-Beltrami operators on M and ∂M , respectively, and consider the eigenvalue problem for Wentzell boundary conditions ∆u = 0 in M, −β∆u + ∂ ν u = λu on ∂M, (1.1) where β is a given real number and ∂ ν denotes the outward unit normal derivative. When M is a bounded domain in a Euclidean space, the above problem has been studied recently in [7]. A general derivation of Wentzell boundary conditions can be found in [13].…”

Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues

“…Obviously, if b = 0, then, we recover the classical Steklov problem. The spectrum of this problem is an increasing sequence (see [15])…”

“…The eigenvalue 0 is simple and the corresponding eigeinfunctions are the constant ones. Moreover (see [15,37]), α 1 has the following variational characterization…”