The Size of the First Eigenfunction of a Convex Planar Domain

The goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains. In particular, we will locate the place where u achieves its maximum to within a distance comparable to the inradius, uniformly for arbitrarily large diameter. In addition, we will estimate the location of other level sets of u by showing that u is well-approximated by the first eigenfunction of a naturally associated ordinary differential (Schrödinger) operator. We intend to show in a separate paper that the estimates here are best possible in order of magnitude. The present paper depends on the ideas and results of our earlier work [J] and [GJ], where detailed estimates for the zero set of the second eigenfunction (or first nodal line) are obtained. The paper [J] also contains some estimates for the first eigenfunction and lowest eigenvalue, but the techniques of [GJ] and new techniques introduced here are essential to the best possible estimates for the first eigenfunction presented here. The maximum of the first eigenfunction occurs at the point of largest displacement of a vibrating drum with fixed edges when it vibrates at its fundamental or first resonant frequency. The first nodal line is the stationary curve of the drum at the second resonant frequency. The maximum is harder to find experimentally than the nodal line because it is a single point. Its location has less influence on the eigenvalue or Dirichlet integral, so it is also harder to locate mathematically. Another way to describe the difficulty is as follows. To find the maximum of the first eigenfunction we will need to estimate its first directional derivative. Derivatives of the first eigenfunction are, roughly speaking, analogous to the second eigenfunction because they are solutions to an eigenfunction equation. Moreover, convexity properties of the first eigenfunction imply that the zero set of the derivative divides the region into two connected components. But the derivatives are harder to estimate than a second eigenfunction because they do not vanish at the boundary.