Thin triangles

The purpose of this page is to establish

Theorem: the hot spots conjecture is true for acute-angled triangles ABC when [math]\displaystyle{ \angle BAC }[/math] is sufficiently small.

Let us write [math]\displaystyle{ \varepsilon := \angle BAC }[/math], which we view as being small. We may normalize A = (0,0) and B = (1,0). Since the other two angles [math]\displaystyle{ \angle ABC, ACB }[/math] are [math]\displaystyle{ \pi/2 - O(\varepsilon) }[/math], we may also normalize [math]\displaystyle{ C = (1 + O(\varepsilon^2), \varepsilon + O(\varepsilon^2)) }[/math].

For the sector of radius 1 and aperture [math]\displaystyle{ \varepsilon }[/math], we know that the second eigenvalue is [math]\displaystyle{ j_1^{-2} }[/math], where [math]\displaystyle{ j_1 }[/math] is the first solution to [math]\displaystyle{ J'_0(j_1)=0 }[/math] (or equivalently [math]\displaystyle{ J_1(j_1)=0 }[/math]).