On the ground state eigenfunction of a convex domain in Euclidean space

Abstract

We study the first eigenfunction φ ₁ of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that $\left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2 \to \infty$ if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that $(vol(D))^{1/2} \left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2$ is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D≈ of D with given volume.