Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

A Sharp Upper Bound for the First Dirichlet Eigenvalue and the Growth of the Isoperimetric Constant of Convex Domains

We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Pólya and Szegö’s 1951 upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains...

On the First Eigenvalue of a Fourth Order Steklov Problem

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 problem...

The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization

We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure. Mathematics Subject Classification (2000)....

The application of isoperimetric inequalities for nonlinear eigenvalue problems

Our aim is to show the interplay between geometry analysis and applications of the theory of isoperimetric inequalities for some nonlinear problems. Reviewing the isoperimetric inequalities valid on Minkowskian plane we show that we can get estimations of physical quantities, namely, estimation on the first eigenvalue of nonlinear eigenvalue problems, on the basis of easily accessible geometric...

MRRW Bound and Isoperimetric Problems