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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 .

Isoperimetric and universal inequalities for eigenvalues