Hot Spots Problem Illustrated

A painting by Ellsworth Kelly illustrates the following problem.

Is it true that the second Neumann eigenfunction in a triangle with acute angles attains its maximum and minimum on the boundary of the triangle?

The problem has been solved by Chris Judge and Sugata Mondal. See "Euclidean triangles have no hot spots." Annals of Mathematics, 191(1):167–211, 2020; "Erratum: Euclidean triangles have no hot spots." Annals of Mathematics, 195(1):337–362, 2022.

The green and red areas in the painting are conjectured to represent parts of the triangle where the second Neumann eigenfunction is negative and postive.


 

Ellsworth Kelly Two Panels: Green Orange (1970)
Oil on canvas, Carnegie Musuem of Art, Pittsburgh

For more information on the "hot spots" problem, see

• K. Burdzy and R. Bañuelos, On the "hot spots" conjecture of J. Rauch J. Func. Anal. 164 (1999) 1-33
• K. Burdzy and W. Werner, A counterexample to the "hot spots" conjecture Ann. Math. 149 (1999) 309-317
• R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains Journal AMS 17 (2004) 243-265
• K. Burdzy, Neumann eigenfunctions and Brownian couplings Potential theory in Matsue. Proceedings of the International Workshop on Potential Theory, Matsue 2004. Advanced Studies in Pure Mathematics 44. Mathematical Society of Japan, 2006, pp. 11-23 (review paper)

A significant new development has been announced in the preprint arXiv:2412.06344 by Jaume de Dios Pont "Convex sets can have interior hot spots."