Courant Sharp Eigenvalues And K-minimal Partitions Of Dirchlet Laplacian

This paper focuses on the relation of k-minimal partitions and Courant sharp eigen-values.Many mathematicians pay attention to the progress in k-minimal partitions,such as P.Berard,B.Helffer and F.Lin.A k-minimal partition shows when a domain is of minimal energy if it is divided into k pieces.However,it is not easy to determine wether a partition is minimal,even if the 3-minimal partition of the unit disc.Fortunately,par-titions produced by Courant sharp eigenfunctions provide some examples.In this paper,we will introduce the definition and some basic properties of Courant sharp eigenvalues and k-minimal partitions.Several examples will also be presented.Nodal partitions of Courant sharp eigenfunctions are always minimal partitions but the converse is not true.We will give some suffice conditions to decide when a k-minimal partition is also a n-odal partition,which is the main theorem of the paper.At the last of the paper,some examples of k-minimal partitions will be shown,with k small.