Mark S. Ashbaugh

Spectral Theory and Geometry: Isoperimetric and universal inequalities for eigenvalues

Abstract 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 present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H.C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S n or H n . Introduction The Eigenvalue Problems The first eigenvalue problem we shall introduce is that of the fixed membrane, or Dirichlet Laplacian . We consider the eigenvalues and eigenfunctions of –Δ on a bounded domain (=connected open set) Ω in Euclidean space R n , i.e., the problem It is well-known that this problem has a real and purely discrete spectrum where Here each eigenvalue is repeated according to its multiplicity. An associated orthonormal basis of real eigenfunctions will be denoted u 1 , u 2 , u 3 , …. In fact, throughout this paper we will assume that all functions we consider are real. This entails no loss of generality in the present context. The next problem we introduce is that of the free membrane, or Neumann Laplacian .

More bounds on eigenvalue ratios for Dirichlet Laplacians in N dimensions

The authors investigate bounds for various combinations of the low eigenvalues of the Laplacian with Dirichlet boundary conditions on a bounded domain

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

\Omega \subset \mathbb{R}^n

Universal bounds for the low eigenvalues of Neumann Laplacians in N dimensions

. These investigations continue and expand upon earlier work of Payne, Polya, Weinberger, Brands, Chiti, and the authors of this present paper. In particular, the authors generalize and extend to the n-dimensional setting various bounds of Payne, Polya, Weinberger, Brands, and Chiti and examine their consequences and interrelationships in detail. This includes comparing the asymptotic forms of the various bounds as the dimension n becomes large. The authors also present various extensions and consequences of their recent proof of the Payne–Polya–Weinberger conjecture, including the proof of a second conjecture of Payne, Polya, and Weinberger under an added symmetry condition.

Optimal Bounds for Ratios of Eigenvalues of One-Dimensional Schrodinger Operators with Dirichlet Boundary Conditions and Positive Potentials

Abstract We study spectral properties for H K , Ω , the Krein–von Neumann extension of the perturbed Laplacian − Δ + V defined on C 0 ∞ ( Ω ) , where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ R n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C 1 , r , r > 1 / 2 . In particular, in the aforementioned context we establish the Weyl asymptotic formula # { j ∈ N | λ K , Ω , j ⩽ λ } = ( 2 π ) − n v n | Ω | λ n / 2 + O ( λ ( n − ( 1 / 2 ) ) / 2 ) as λ → ∞ , where v n = π n / 2 / Γ ( ( n / 2 ) + 1 ) denotes the volume of the unit ball in R n , and λ K , Ω , j , j ∈ N , are the non-zero eigenvalues of H K , 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 − Δ + 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 1980s. Our work builds on that of Grubb in the early 1980s, 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 = R n ∖ K , n ⩾ 3 , with K ⊂ R n compact and vanishing Bessel capacity B 2 , 2 ( K ) = 0 , to prove equality of Friedrichs and Krein Laplacians in L 2 ( Ω ; d n x ) , that is, − Δ | C 0 ∞ ( Ω ) has a unique nonnegative self-adjoint extension in L 2 ( Ω ; d n x ) .

Perturbation theory for shape resonances and large barrier potentials

The authors consider bounds on the Neumann eigenvalues of the Laplacian on domains in

A second proof of the Payne-Pólya-Weinberger conjecture

I\mathbb{R}^n

Open Problems on Eigenvalues of the Laplacian

in the light of their recent results on Dirichlet eigenvalues, in particular, their proof of t...

Maximal and minimal eigenvalues and their associated nonlinear equations

AbstractConsider the Schrödinger equation −u″+V(x)u=λu on the intervalI⊂ℝ, whereV(x)≧0 forx∈I and where Dirichlet boundary conditions are imposed at the endpoints ofI. We prove the optimal bound