7 Spectral inequalities in quantitative form

Abstract

We review some results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian. 27 4.2. A two-dimensional result by Nadirashvili 29 4.3. The Szegő-Weinberger inequality in sharp quantitative form 32 4.4. Checking the sharpness 35 5. Stability for the Brock-Weinstock inequality 40 5.1. A quick overview of the Steklov spectrum 40 5.2. Weighted perimeters 41 5.3. The Brock-Weinstock inequality in sharp quantitative form 43 5.4. Checking the sharpness 45 6. Some further stability results 46 6.

Surgery principles for the spectral analysis of quantum graphs

“…Proposition 6.6 and Corollary 6.7 for more details. Theorems 6.3 and 6.5 may be viewed as quantitative isoperimetric inequalities, which make an appearance in spectral geometry of domains in higher dimensions [BDP17]. Such inequalities give not just a sharp bound on an eigenvalue in terms of the total volume (or in our case length of the graph), but also a correction term which takes into account some measure of the difference of a given domain from the optimising one (the size of the doubly connected component or the length of the longest cycle, in our case).…”

The quantitative Faber–Krahn inequality for the Robin Laplacian

“…The recent survey by Brasco and De Philippis [4] gives a complete and up to date overview of the topic. The quantitative estimate for the Robin eigenvalue is left as an open problem, and the purpose of our paper is to solve it.…”

Stability of the Bakry-Émery theorem on Rn

“…The function g is measurable and satisfies g Lip ≤ √ π, as already shown in (9). Hence, we integrate with respect to µ to conclude f dµ − f dγ 0,e dμ = yg(y, z) − ∂ y g(y, z) dµ…”

“…Poincaré inequalities can be viewed as estimates on the smallest eigenvalue of the diffusion operator −∆ + ∇V · ∇. Stability for other spectral problems have been considered, such as Poincaré inequalities on bounded domains [8,9] and a lower bound on the spectrum of Schrödinger operators [13], respectively with applications in shape optimization and quantum mechanics.…”