Original Paper Area 1

Japan Journal of Industrial and Applied Mathematics

, Volume 30, Issue 1, pp 145-164

First online:

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

  • Yasuhito MiyamotoAffiliated withDepartment of Mathematics, Faculty of Science and Technology, Keio University Email author 

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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.


Hot spots Convex domain Nodal line Maximum point

Mathematics Subject Classification (2010)

Primary 35J25 35B38 Secondary 35J05 35P15