Polymath7 research threads 4: the Hot Spots Conjecture

It’s time for another rollover of the  Polymath7 “Hot Spots” conjecture, as the previous research thread has again become full.

Activity has now focused on a numerical strategy to solve the hot spots conjecture for all acute angle triangles ABC.  In broad terms, the strategy (also outlined in this document) is as follows.   (I’ll focus here on the problem of estimating the eigenfunction; one also needs to simultaneously obtain control on the eigenvalue, but this seems to be to be a somewhat more tractable problem.)

1. First, observe that as the conjecture is scale invariant, the only relevant parameters for the triangle ABC are the angles , which of course lie between 0 and and add up to .  We can also order , giving a parameter space which is a triangle between the values .
2. The triangles that are too close to the degenerate isosceles triangle or the equilateral triangle need to be handled by analytic arguments.  (Preliminary versions of these arguments can be found here and Section 6 of these notes  respectively, but the constants need to be made explicit (and as strong as possible)).
3. For the remaining parameter space, we will use a sufficiently fine discrete mesh of angles ; the optimal spacing of this mesh is yet to be determined.
4. For each triplet of angles in this mesh,  we partition the triangle ABC (possibly after rescaling it to a reference triangle , such as the unit right-angled triangle) into smaller subtriangles, and approximate the second eigenfunction (or the rescaled triangle ) by the eigenfunction (or ) for a finite element restriction of the eigenvalue problem, in which the function is continuous and piecewise polynomial of low degree (probably linear or quadratic) in each subtriangle; see Section 2.2 of these notes.    With respect to a suitable basis, can be represented by a finite vector .
5. Using numerical linear algebra methods (such as Lanczos iteration) with interval arithmetic, obtain an approximation to , with rigorous bounds on the error between the two.  This gives an approximation to or with rigorous error bounds (initially of L^2 type, but presumably upgradable).
6. After (somehow) obtaining a rigorous error bound between and (or and ), conclude that stays far from its extremum when one is sufficiently far away from the vertices A,B,C of the triangle.
7. Using stability theory of eigenfunctions (see Section 5 of these notes), conclude that stays far from its extremum even when is not at a mesh point.  Thus, the hot spots conjecture is not violated away from the vertices.  (This argument should also handle the vertex that is neither the maximum nor minimum value for the eigenfunction, leaving only the neighbourhoods of the two extremising vertices to deal with.)
8. Finally, use an analytic argument (perhaps based on these calculations) to show that the hot spots conjecture is also not violated near an extremising vertex.

This all looks like it should work in principle, but it is a substantial amount of effort; there is probably still some scope to try to simplify the scheme before we really push for implementing it.