A New Approach to the Hot Spots Conjecture

Explore a new approach to the hot spots conjecture in this 28-minute lecture from the Workshop on "Spectral Theory of Differential Operators in Quantum Theory" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the history of the conjecture, dating back to J. Rauch in 1974, which posits that the hottest and coldest spots in an insulated homogeneous medium should converge to the boundary over time. Examine the alternative formulation involving eigenfunctions of the Neumann Laplacian on Euclidean domains. Learn about recent advances, including the proof for all triangles by Judge and Mondal in 2020, and understand why the conjecture remains open for simply connected or convex domains. Discover a novel approach based on a non-standard variational principle for eigenvalues of Neumann and Dirichlet Laplacians, potentially offering new insights into this long-standing mathematical problem.