Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.

A sharp trace inequality for functions of bounded variation in the ball

The best constant in a mean-value trace inequality for functions of bounded variation on admissible domains Ω ⊂ ℝn is shown to agree with an isoperimetric constant associated with Ω. The existence and form of extremals is also discussed. This result is exploited to compute the best constant in the relevant trace inequality when Ω is a ball. The existence and the form of extremals in this special case turn out to depend on the dimension n. In particular, the best constant is not achieved when Ω is a disc in ℝ2.

On the Cheeger–Buser Type Inequalities for the p-Laplacian

We study the relationship between the first eigenvalue of the p-Laplacian (p > 1) and Cheeger’s isoperimetric constant for families of compact manifolds and graphs with the Cheeger constant converging to zero.