ON THE GROUND-STATE EIGENFUNCTION OF A CONVEX DOMAIN IN EUCLIDEAN-SPACE

We study the first eigenfunction phi(1) of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel fu nctions yield that parallel to phi(1) parallel to(infinity)/parallel t o phi(1) parallel to(2) --> infinity if D is a sector in Euclidean pla ne with area 1 and the angle tends to 0. We aim to characterize those domains D such that (vol(D))(1/2)parallel to phi(1) parallel to(infini ty)/parallel to phi(1) parallel to(2) is large in terms of the ratio o f the first eigenvalue of D and the infimum of the first eigenvalues o f all subdomains (D) over tilde of D with given volume.