Showing papers by "Steven J. Cox published in 1993"
01 Jan 1993
TL;DR: In this paper, the eigenvalues of the Laplacian subject to Dirichlet boundary conditions over star-like planar sets of bounded perimeter and prescribed area are extremized, and necessary conditions are obtained via a marriage of Kato's perturbation theory and Clarke's nonsmooth calculus.
Abstract: We extremize the eigenvalues of the Laplacian subject to Dirichlet boundary conditions over starlike planar sets of bounded perimeter and prescribed area. Existence of such extremizers follows from Caratheodory’s notion of set convergence while the necessary conditions are obtained via a marriage of Kato’s perturbation theory and Clarke’s nonsmooth calculus. This necessary condition implies that where the boundary of the extremizer possesses a Holder continuous second derivative it is in fact infinitely differentiable.