The “hot spots” conjecture for a certain class of planar convex domains

Abstract

We prove the “hot spots” conjecture of Rauch [“Five problems: An introduction to the qualitative theory of partial differential equations,” Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics (Springer, Berlin, 1975), Vol. 446, pp. 355–369] for a certain class of planar convex domains. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Ω attains its maximum (minimum) at points in ∂Ω. One class of domains is the planar convex domain Ω satisfying diam(Ω)2/|Ω|<1.378. When Ω is a disk, diam(Ω)2/|Ω|≈1.273. Hence, this condition indicates that Ω is a nearly circular planar convex domain. However, symmetries of the domain are not assumed. We give other sufficient conditions for domains for which the conjecture holds. We also give a new isoperimetric inequality.