Japan Journal of Industrial and Applied Mathematics

, Volume 30, Issue 1, pp 145–164

A planar convex domain with many isolated “ hot spots” on the boundary

Authors

    • Department of Mathematics, Faculty of Science and TechnologyKeio University
Original Paper Area 1

DOI: 10.1007/s13160-012-0091-z

Cite this article as:
Miyamoto, Y. Japan J. Indust. Appl. Math. (2013) 30: 145. doi:10.1007/s13160-012-0091-z

Abstract

We construct a convex domain such that the second Neumann eigenfunction has an arbitrary number of isolated local maximum points on the boundary. This domain, which is close to a sector, is constructed by combining thin isosceles triangles. Then we study the shape of the second Neumann eigenfunction on isosceles triangles. In particular, we show that if the isosceles triangle is subequilateral, then the second Neumann eigenvalue is simple and the associated eigenfunction has exactly two maximum points which are located at two corners.

Keywords

Hot spots Convex domain Nodal line Maximum point

Mathematics Subject Classification (2010)

Primary 35J25 35B38 Secondary 35J05 35P15

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2012