The size of the first eigenfunction of a convex planar domain

Abstract

This paper estimates the size of the first Dirichlet eigenfunction of a convex planar domain. The eigenfunction is shown to be well-approximated, uniformly for all convex domains, by the first Dirichlet eigenfunction of a naturally associated ordinary differential (Schrödinger) operator. In particular, the place where the eigenfunction attains its maximum is located to within a distance comparable to the inradius.