On the ``hot spots'' conjecture of J. Rauch
The conjecture in the title states roughly, that for most initial data \(u_0\) which are subject to the heat equation under no flux on the boundary of a bounded open connected subset of \(\mathbb{R}^n\), the hot spot moves eventually to the boundary. If one uses separations of variables, this movement of the hot spot can be explained if the (or a) first nonconstant eigenfunction of the Laplacian under Neumann boundary conditions attains its maximum (or maximum modulus) on the boundary (only). The brackets indicate slightly different variants. The authors considerably generalize earlier results of the reviewer [Lecture Notes in Mathematics, 1150. Berlin-Springer (1985; Zbl 0593.35002)] and give various sufficient conditions which imply the monotonicity of the eigenfunctions in certain directions and thus the conjecture. If a domain is long and thin (and this is stated in a precise way) for instance, the eigenfunction is monotone in the long direction. The proofs are partly probabilistic and the paper offers a lot of results, insight and new conjectures about the shape of these eigenfunctions.
scientific article; zbMATH DE number 1310878
scientific article