The purpose of this page is to establish
- Theorem: the hot spots conjecture is true for acute-angled triangles ABC when is sufficiently small.
Let us write , which we view as being small. We may normalize A = (0,0) and B = (1,0). Since the other two angles are , we may also normalize .
For the sector of radius 1 and aperture , we know (as discussed at the hot spots conjecture) that the second eigenvalue is , where is the first solution to J'0(j1) = 0 (or equivalently J1(j1) = 0). We claim that for the triangle ABC, the second eigenvalue is .
First, the upper bound. We can take the second eigenfunction for an inscribed sector to ABC of radius and aperture , with , and extend it smoothly to a function on ABC obeying the Neumann boundary conditions. It will not quite mean zero, but we can subtract a constant of size to make it of mean zero. The Rayleigh quotient for this object can be shown to be , providing the upper bound.
The lower bound can be proven by similar methods once we get good enough C^2 type bounds on eigenfunctions on ABC, but for now let us just establish the weaker bound . It suffices to establish the Poincare inequality
for all u on ABC (not necessarily obeying the Neumann boundary condition) and some constant c depending on u. But one can rescale ABC to, say, the unit equilateral triangle in which case the bound is classical, and note that the constants for the rescaling are favorable. (One can also cite here the bounds from [PW1960] or [LS2009] for this bound; the [LS2009] bound in fact gives the lower bound of directly.)
Regularity of eigenfunctions
Let u be the second eigenfunction on ABC, normalised so that
|∫||| u | 2 = | ABC | .|
By repeated integration by parts (as discussed at the hot spots conjecture) we have
for j=0,1,2,3. In particular, is O(1) on the average on ABC for j=0,1,2,3. In the middle third of the triangle, we can use elliptic regularity (after reflecting the triangle across AB and AC many times so that a disk of radius ~1 sits inside the domain) to in fact conclude that all derivatives are O(1) in this region.
By working in polar coordinates around A, we may expand
for some coefficients ck. Because all derivatives of u are O(1) in the middle third of the triangle, we see that is rapidly decreasing in k in this middle third, which from the asymptotics of Bessel functions gives excellent regularity on the left third of the triangle; in particular it is not difficult to get uniform bounds on the C^3 norm in this third.
The situation is more delicate on the right third of the triangle. We begin in the region of size around the edge BC. From (*) we know that on average in this region, while from the Poincare inequality (or Sobolev inequality) one can show that on average in this region. By using Bessel function expansions around B and C one can then show that pointwise here; in the rest of the triangle one can use elliptic regularity to also get first on average in regions of diameter , and then pointwise by elliptic regularity (and reflection). Thus u is bounded uniformly in C^2.
This is already enough regularity on u to extend u to a circumscribing sector and show that . Now we can relate u more carefully with a Bessel function. We again work in polar coordinates around the origin A. If we write
for the averaged radial component of u, defined for (say), then from integration by parts we see that
thus u_0 is a multiple of the Bessel function (recall that u_0 has to be continuous at the origin). From the uniform C^2 bounds, we have for all and . Given the normalisation of u, this implies that
which implies that where c * is the positive absolute constant
Without loss of generality we may take the positive sign, thus , and so we have the asymptotic
throughout the triangle. This is already enough to show that the maximum can only occur within of A, and the minimum can only occur within of B or C.
Now let us show that the maximum can only occur at A. If for contradiction there is a nearby point P to A which also attains the maximum, then at P, but also at A. Hence, by Rolle's theorem, one has at some intermediate point Q on the interval AP. But from the Bessel expansion we see that is bounded away from zero at A, and so from the uniform C^3 bounds we obtain a contradiction if is small enough.
It remains to show that that the minimum can only occur on BC. Let n be the unit normal to BC, thus on BC. If for contradiction there is a nearby point Q to BC which also attains the minimum, then at P, so by Rolle's theorem we have for some intermediate point Q on the line segment from P to BC in the direction n. As n differs from the radial direction by , we conclude from the uniform C^2 bounds that
Large double radial derivative
We need to derive a contradiction from (***). To do this, we begin by inserting (***) into the equation
which is the eigenfunction equation in polar coordinates. As Q is near to BC, we see from (**) that u is comparable to 1, so λu is comparable to 1 also. From the Neumann boundary condition we have on BC, and so since Q is within of BC we conclude from the uniform C^2 bounds that at Q. Putting all this together, we see that
Suppose first that Q is further than from BC. On the circular arc in ABC centered at A and passing through Q, we know that on the ends of this arc, and so by Rolle's theorem we have somewhere in the interior of this arc. In particular we see that at some point R within of Q.
By the mean value theorem, we can find a point S on QR where . From the geometry of the situation, we see that S is a distance at least from B or C. By (*), we thus have on average on a ball of radius centered at S (extending u by reflection as necessary), which implies by elliptic regularity that at S, a contradiction.
Thus we may assume that Q is within of BC. To get a contradiction, we will now need a rather delicate analysis of the eigenfunction u in the neighbourhood of BC.
We first look at what u is doing in the neighbourhood of B. We perform a Bessel expansion in polar coordinates around B, obtaining an expansion of the form
where β is the angle at B, r = rB is now the distance to B, and θ is the angle subtended at B relative to the ray BA. Note that . Evaluating at B we see that c0 = O(1). Since pointwise, and in particular when r is comparable to , we conclude using Bessel function asymptotics that
(say) for and k positive. Since on average in this region by (*), a similar argument gives the improvement
for k > 1 (in order to keep the exponent πk / β well away from 2). From these asymptotics, we see in particular that we can bound the Hessian here by
in the -neighbourhood of B for some bounded constants a0,a1 which are scalar multiples of c0,c1 respectively. Similarly we will have
in the -neighbourhood of C for some bounded constants a'0,a'1, where r_C is now the distance to C, and γ is the angle at C. Comparing these two asymptotics on the common domain, we see that and .
Next, recall that on any circular arc in ABC centered at A, the function has mean zero. Using an arc tangent to BC, and noting from the Neumann boundary conditions and the uniform C^2 bounds that on this arc, we see that has mean on such an arc. Comparing this with the above two asymptotics, we see that . On the other hand, evaluating the Hessian of u at B or at C we see that a0,a'0 are comparable to 1. Putting this together, we see that will be comparable to 1 in the neighbourhood of BC, and in particular at Q, giving the desired contradiction.