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

摘要

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.