On reverse Faber-Krahn inequalities

Abstract

Payne-Weinberger showed that 'among the class of membranes with given area A, free along the interior boundaries and fixed along the outer boundary of given length L 0 , the annulus Ω # has the highest fundamental frequency,' where Ω # is a concentric annulus with the same area as Ω and the same outer boundary length as L 0 .We extend this result for the higher dimensional domains and p-Laplacian with p ∈ (1, ∞), under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of p-Laplacian (with mixed boundary conditions) on a ball and a concentric annulus cannot be a concentric sphere.Keywords Reverse Faber-Krahn inequality · Interior parallels · Isoperimetric inequality · Nagy's inequalities · First eigenvalue of p-Laplacian · Mixed boundary value problems · Elasticity problems · Non-radiality IntroductionLet us first recall the famous conjecture by Lord Rayleigh from his book 'The theory of sound' [23] published in 1877. He conjectured that 'among all planar domains Ω of fixed area, the disk is the domain that minimises the first Dirichlet eigenvalue λ 1 (Ω) of the Laplacian.' This conjecture was open for a very long time. The first proof of this conjecture was published in 1923 by Faber [8]. In 1925, Krahn [14] gave an independent proof for this conjecture, and later he extended the same to the higher dimension. Now this result is collectively known as the Faber-Krahn inequality, and it states that:with the equality if and only if Ω is a ball (up to a measure zero set), where Ω * is a ball of the same measure as Ω.For more on Faber-Krahn inequality and related results, we refer to [18] and [21].

Optimal Shapes for the First Dirichlet Eigenvalue of the $p$-Laplacian and Dihedral symmetry

“…However, the strict monotonicity has been obtained by Anoop, Bobkov, and Sasi in [8]. The results related to the optimization of eigenvalues with respect to other boundary conditions can be found in [5][6][7]. For further literature and open problems in this direction, we refer to the book [23]; see also [22] for recent developments in spectral geometry.…”

Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigenvalue

“…The main ideas consist in constructing judicious test functions by using the notion of web-functions (see [15] for more details on web functions). These ideas were very recently used and adapted for other similar problems (see [4,42]). A classical family of obstacle problems that attracted a lot of attention was to find the best emplacement of a spherical hole inside a ball that optimizes the value of a given spectral functional (see [6], section (9)).…”

Szegő-Weinberger type inequalities for symmetric domains with holes

“…The proof of Lemma 2.3 is placed in Appendix A. Notice that the weaker inequality 𝜇 1,1 < 𝜇 0,2 can be obtained from [2,Theorems 1.4] or [20,Theorem 1.2], see also Proposition A.2 for a simple proof. However, (2.12) cannot be improved, in general, to 𝜇 3,1 < 𝜇 0,2 .…”