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A variational approach to the hot spots conjecture (2404.01890v1)
Published 2 Apr 2024 in math.SP, math-ph, math.AP, and math.MP
Abstract: We review a recent new approach to the study of critical points of Laplacian eigenfunctions. Its core novelty is a non-standard variational principle for the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, simply connected planar domains. This principle can be used to provide simple proofs of some previously known results on the hot spots conjecture.
- R. Bañuelos and K. Burdzy, On the “hot spots” conjecture of J. Rauch, J. Funct. Anal. 164 (1999), 1–33.
- R. F. Bass and K. Burdzy, Fiber Brownian motion and the “hot spots” problem, Duke Math. J. 105 (2000), no. 1, 25–58.
- K. Burdzy and W. Werner, A counterexample to the “hot spots” conjecture, Ann. of Math. (2) 149 (1999), 309–317.
- D. Jerison and N. Nadirashvili, The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc. 13 (2000), no. 4, 741–772.
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