校级学术报告
报告题目: On Nodal Domains and Spectral Minimal Partitions
演讲人： Professor Bernard Helffer
University of Paris 11
时间： 2008年4月29日（星期二）下午2：00-3：00
地点： 闵行校区3＃330教室
摘要：Given an open set and a partition of it by k open sets ωj , we can consider the quantity maxj λ(ωj ) where λ(ωj ) is the ground state energy of the Dirichlet realization of the Laplacian in ωj . If we denote by L(k) the infimum over all the k-partitions of maxj λ(ωj ), a minimal k-partition is then a partition which realizes the infimum. Although the analysis is rather standard when k = 2 (we find the nodal domains of a second eigenfunction), the analysis of higher k’s becomes non trivial and quite interesting. In this talk, we would like to discuss the properties of minimal spectral partitions, illustrate the difficulties by considering simple cases like the disc or the square (k = 3) and will also exhibit the possible role of the hexagone in the asymptotic behavior as k → +∞ of L(k). This work has started in collaboration with T. Hoffmann-Ostenhof and has been continued (published or in progress) in collaboration with (by alphabetic order) V. Bonnaillie-Noel, T. Hoffmann-Ostenhof, S. Terracini, G. Verzini, and G. Vial.