A planar convex domain with many isolated “ hot spots” on the boundary
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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.
KeywordsHot spots Convex domain Nodal line Maximum point
Mathematics Subject Classification (2010)Primary 35J25 35B38 Secondary 35J05 35P15
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The author thanks anonymous referees for the careful reading of the manuscript and comments pointing out that the proofs of Lemmas 3.2 and 3.5 of the previous version of the manuscript were imperfect.
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