On principal frequencies and inradius in convex sets

Abstract

We generalize to the case of the p-Laplacian an old result by Hersch and Protter. Namely, we show that it is possible to estimate from below the first eigenvalue of the Dirichlet p-Laplacian of a convex set in terms of its inradius. We also prove a lower bound in terms of isoperimetric ratios and we briefly discuss the more general case of Poincarè-Sobolev embedding constants. Eventually, we highlight an open problem.

Keywords

• convex sets
• p-laplacian
• nonlinear eigenvalue problems
• inradius
• cheeger constant