Hot Spots Problem Illustrated
A painting by Ellsworth Kelly illustrates
the following open 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 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
Oil on canvas,
Carnegie Musuem of Art, Pittsburgh
For more information on the "hot spots"
- 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,