Payne-Pólya-Weinberger conjecture, Sperner’s inequality, biharmonic operator, bi-Laplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the Pólya-Szegő conjecture, universal inequalities for eigenvalues, Hile-Protter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round. We also comment on the use of coordinate transformations for these operators and mention some open problems.

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z): = � (z − λk) σ +.

Abstract. For a domain Ω contained in a hemisphere of the n–dimensional sphere Sn we prove the optimal result λ2/λ1(Ω) ≤ λ2/λ1(Ω ⋆)fortheratio of its first two Dirichlet eigenvalues where Ω ⋆ , the symmetric rearrangement of Ω in Sn, is a geodesic ball in Sn having the same n–volume as Ω. We also show that λ2/λ1 for geodesic balls of geodesic radius θ1 less than or equal to π/2 is an increasing function of θ1 which runs between the value (jn/2,1/jn/2−1,1) 2 for θ1 = 0 (this is the Euclidean value) and 2(n +1)/n for θ1 = π/2. Here jν,k denotes the kth positive zero of the Bessel function Jν(t). This result generalizes the Payne–Pólya–Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of Sn and having a fixed value of λ1 the one with the maximal value of λ2 is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for λ2/λ1. Various other results for λ1 and λ2 of geodesic balls in Sn are proved in the course of our work. 1.

Abstract. We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N | λK,Ω,j ≤ λ} = (2π) −n vn|Ω | λ n/2 + O ( λ (n−(1/2))/2) as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the non-zero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exterior-type domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In N dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and γ lower semicontinuous.

Abstract. We prove that the Hersch–Payne–Schiffer isoperimetric inequality for the n-th nonzero Steklov eigenvalue of a bounded simply– connected planar domain is sharp for all n ≥ 1. The equality is attained in the limit by a sequence of simply–connected domains degenerating to the disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch–Payne–Schiffer inequality for n = 2 and show that it is strict in this case. 1. Introduction and

Abstract. We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Pólya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres. 1. Introduction and

Abstract. We prove some results about the first Steklov eigenvalue d1 of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality [9] may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d1 for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball. 1.