OAI.StrictHotSpots.main_theorem

The theorem states that, for a set Ω in the Euclidean plane ℝ² that is admissible, meaning nonempty, open, bounded, simply connected and with smooth boundary (near each boundary point p there is an open neighborhood U and a C^∞ function ρ on U with ρ(p)=0, nonzero derivative at p, and Ω∩U={ρ<0}∩U), and for any function u on the plane lying in the first Neumann eigenspace of Ω and not identically zero on Ω, the conclusion MainConclusion holds. Membership in the first Neumann eigenspace means that u is C^∞ on the closure of Ω, u and its gradient (the plane gradient of u) are square-integrable on Ω with the gradient being the weak gradient of u on Ω, the integral of u over Ω is zero, and for every square-integrable v with square-integrable weak gradient g on Ω, the integral over Ω of ⟨∇u, g⟩ equals λ times the integral over Ω of u·v. Here λ is the first positive Neumann value, defined as the infimum of the Rayleigh quotients ∫‖g‖²/∫v² over all such pairs (v,g) with ∫v=0 and ∫v²>0. The conclusion MainConclusion has two parts: the gradient of u is nonzero at every point of Ω, and for every x in Ω, u(x) lies strictly between the infimum and the supremum of u over the boundary of Ω.