The polymath blog

It’s once again time to roll over the research thread for the Polymath7 “Hot Spots” conjecture, as the previous research thread has again become full.

As the project moves into a more mature stage, with most of the “low-hanging fruit” already collected, progress is now a bit less hectic, but our understanding of the problem is improving.  For instance, in the previous thread, the relationship between two different types of arguments to obtain monotonicity of eigenfunctions – namely the coupled Brownian motion methods of Banuelos and Burdzy, and an alternate argument based on vector-valued maximum principles – was extensively discussed, and it is now fairly clear that the two methods yield a more or less equivalent set of results (e.g. monotonicity for obtuse triangles with Neumann conditions, or acute triangles with two Neumann and one Dirichlet side).  Unfortunately, for scalene triangles it was observed that the behaviour of eigenfunctions near all three vertices basically preclude any reasonable monotonicity property from taking place (in particular, the conjecture in the preceding thread in this regard was false).  This is something of a setback, but perhaps there is some other monotonicity-like property which could still hold for scalene acute triangles, and which would imply the hot spots conjecture for these triangles.  We do now have a number of accurate numerical representations of eigenfunctions, as well as some theoretical understanding of their behaviour, especially near vertices or near better-understood triangles (such as the equilateral or isosceles right-angled triangle) so perhaps they could be used to explore some of these properties.

Another result claimed in previous threads – namely, a theoretical proof of the simplicity of the second eigenvalue – has now been completed, with fairly good bounds on the spectral gap, which looks like a useful thing to have.

It was realised that a better understanding of the geometry of the nodal line would be quite helpful – in particular its convexity, which would yield the hot spots conjecture on one side of the nodal line at least.   We did establish one partial result in this direction, namely that the nodal line cannot hit the same edge of the triangle twice, but must instead straddle two edges of the triangle (or a vertex and an opposing edge, though presumably this case only occurs for isosceles triangles).  Unfortunately, more control on the nodal line is needed.

In the absence of a definitive theoretical approach to the problem, the other main approach is via rigorous numerics – to obtain, for a sufficiently dense mesh of test triangles, a collection of numerical approximations to second eigenfunctions which are provably close (in a suitable norm) to a true second eigenfunction, and whose extrema (or near-extrema) only occur at vertices (or near-vertices).  In principle, this sort of information would be good enough to rigorous establish the hot spots conjecture for such a test triangle as well as nearby perturbations of that triangle.  The details of this approach, though, are still being worked out.  (And given that they could be a bit messy, it may well be a good idea to not proceed too prematurely with the numerical approach, in case some better approach is discovered in the near future).  One proposal is to focus on a single typical triangle (e.g. the 40-60-80 triangle) as a test case in order to fix parameters.

There was also some further exploration of whether reflection methods could be pushed further.   It was pointed out that even in the very simple case of the unit interval [0,1], it is not obvious (even heuristically) from reflection arguments why the hot spots conjecture should be true.  Reflecting around a vertex whose angle does not go evenly into has created a number of technical difficulties which seem to so far be unsatisfactorily addressed (but perhaps getting an alternate proof of hot spots in the model cases where reflection does work, i.e. the 30-60-90, 45-45-90, and 60-60-60 triangles, would be worthwhile).

The previous research thread for the Polymath7 “Hot Spots Conjecture” project has once again become quite full, so it is again time to roll it over and summarise the progress so far.

Firstly, we can update the map of parameter space from the previous thread to incorporate all the recent progress (including some that has not quite yet been completed):

This map reflects the following new progress:

1. We now have (or will soon have) a rigorous proof of the simplicity of the second Neumann eigenvalue for all non-equilateral acute triangles (this is the dotted region), thus finishing off the first part of the hot spots conjecture in all cases.  The main idea here is to combine upper bounds on the second eigenvalue (obtained by carefully choosing trial functions for the Rayleigh quotient), with lower bounds on the sum of the second and third eigenvalues, obtained by using a variety of lower bounds coming from reference triangles such as the equilateral or isosceles right triangle.  This writeup contains a treatment of those triangles close to the equilateral triangle, and it is expected that the other cases can be handled similarly.
2. For super-equilateral triangles (the yellow edges) it is now known that the extreme points of the second eigenfunction occur at the vertices of the base, by cutting the triangle in half to obtain a mixed Dirichlet-Neumann eigenvalue problem, and then using the synchronous Brownian motion coupling method of Banuelos and Burdzy to show that certain monotonicity properties of solutions to the heat equation are preserved.  This fact can also be established via a vector-valued maximum principle.  Details are on the wiki.
3. Using stability of eigenfunctions and eigenvalues with respect to small perturbations (at least when there is a spectral gap), one can extend the known results for right-angled and non-equilateral triangles to small perturbations of these triangles (the orange region).  For instance, the stability results of Banuelos and Pang already give control of perturbed eigenfunctions in the uniform norm; since for right-angled triangles and non-equilateral triangles, the extrema only occur at vertices, and from Bessel expansion and uniform C^2 bounds we know that for any perturbed eigenfunction, the vertices will still be local extrema at least (with a uniform lower bound on the region in which they are extremisers), we conclude that the global extrema will still only occur at vertices for perturbations.
4. Some variant of this argument should also work for perturbations of the equilateral triangle (the dark blue region).  The idea here is that the second eigenfunction of a perturbed equilateral triangle should still be close (in, say, the uniform norm) to some second eigenfunction of the equilateral triangle.  Understanding the behaviour of eigenfunctions nearly equilateral triangles more precisely seems to be a useful short-term goal to pursue next in this project.

But there is also some progress that is not easily representable on the above map.  It appears that the nodal line of the second eigenfunction may play a key role.  By using reflection arguments and known comparison inequalities between Dirichlet and Neumann eigenvalues, it was shown that the nodal line cannot hit the same edge twice, and thus must straddle two distinct edges (or a vertex and an opposing edge).   (The argument is sketched on the wiki.) If we can show some convexity of the nodal line, this should imply that the vertex straddled by the nodal line is a global extremum by the coupled Brownian motion arguments, and the only extremum on this side of the nodal line, leaving only the other side of the nodal line (with two vertices rather than one) to consider.

We’re now also getting some numerical data on eigenvalues, eigenfunctions, and the spectral gap.  The spectral gap looks reasonably large once one is away from the degenerate triangles and the equilateral triangle, which bodes well for an attempt to resolve the conjecture for acute angled triangles by rigorous numerics and perturbation theory.  The eigenfunctions also look reassuringly monotone in various directions, which suggests perhaps some conjectures to make  in this regard (e.g. are eigenfunctions always monotone along the direction parallel to the longest side?).

This isn’t a complete summary of the discussion thus far – other participants are encouraged to summarise anything else that happened that bears repeating here.

The previous research thread for the Polymath7 project “the Hot Spots Conjecture” is now quite full, so I am now rolling it over to a fresh thread both to summarise the progress thus far, and to make it a bit easier to catch up on the latest developments.

The objective of this project is to prove that for an acute angle triangle ABC, that

1. The second eigenvalue of the Neumann Laplacian is simple (unless ABC is equilateral); and
2. For any second eigenfunction of the Neumann Laplacian, the extremal values of this eigenfunction are only attained on the boundary of the triangle.  (Indeed, numerics suggest that the extrema are only attained at the corners of a side of maximum length.)

To describe the progress so far, it is convenient to draw the following “map” of the parameter space.  Observe that the conjecture is invariant with respect to dilation and rigid motion of the triangle, so the only relevant parameters are the three angles of the triangle.  We can thus represent any such triangle as a point in the region .  The parameter space is then the following two-dimensional triangle:

Thus, for instance

1. A,N,P represent the degenerate obtuse triangles (with two angles zero, and one angle of 180 degrees);
2. B,F,O represent the degenerate acute isosceles triangles (with two angles 90 degrees, and one angle zero);
3. C,E,G,I,L,M represent the various permutations of the 30-60-90 right-angled triangle;
4. D,J,K represent the isosceles right-angled triangles (i.e. the 45-45-90 triangles);
5. H represents the equilateral triangle (i.e. the 60-60-60 triangle);
6. The acute triangles form the interior of the region BFO, with the edges of that region being the right-angled triangles, and the exterior being the obtuse triangles;
7. The isosceles triangles form the three line segments NF, BP, AO.  Sub-equilateral isosceles triangles (with apex angle smaller than 60 degrees) comprise the open line segments BH,FH,OH, while super-equilateral isosceles triangles (with apex angle larger than 60 degrees) comprise the complementary line segments AH, NH, PH.

Of course, one could quotient out by permutations and only work with one sixth of this diagram, such as ABH (or even BDH, if one restricted to the acute case), but I like seeing the symmetry as it makes for a nicer looking figure.

Here’s what we know so far with regards to the hot spots conjecture:

1. For obtuse or right-angled triangles (the blue shaded region in the figure), the monotonicity results of Banuelos and Burdzy show that the second claim of the hot spots conjecture is true for at least one second eigenfunction.
2. For any isosceles non-equilateral triangle, the eigenvalue bounds of Laugesen and Siudeja show that the second eigenvalue is simple (i.e. the first part of the hot spots conjecture), with the second eigenfunction being symmetric around the axis of symmetry for sub-equilateral triangles and anti-symmetric for super-equilateral triangles.
3. As a consequence of the above two facts and a reflection argument found in the previous research thread, this gives the second part of the hot spots conjecture for sub-equilateral triangles (the green line segments in the figure). In this case, the extrema only occur at the vertices.
4. For equilateral triangles (H in the figure), the eigenvalues and eigenfunctions can be computed exactly; the second eigenvalue has multiplicity two, and all eigenfunctions have extrema only at the vertices.
5. For sufficiently thin acute triangles (the purple regions in the figure), the eigenfunctions are almost parallel to the sector eigenfunction given by the zeroth Bessel function; this in particular implies that they are simple (since otherwise there would be a second eigenfunction orthogonal to the sector eigenfunction).  Also, a more complicated argument found in the previous research thread shows in this case that the extrema can only occur either at the pointiest vertex, or on the opposing side.

So, as the figure shows, there has been some progress on the problem, but there are still several regions of parameter space left to eliminate.  It may be possible to use perturbation arguments to extend validity of the hot spots conjecture beyond the known regions by some quantitative extent, and then use numerical verification to finish off the remainder.  (It appears that numerics work well for acute triangles once one has moved away from the degenerate cases B,F,O.)

The figure also suggests some possible places to focus attention on, such as:

1. Super-equilateral acute triangles (the line segments DH, GH, KH).  Here, we know the second eigenfunction is simple (and anti-symmetric).
2. Nearly equilateral triangles (the region near H).  The perturbation theory for the equilateral triangle could be non-trivial due to the repeated eigenvalue here.
3. Nearly isosceles right-angled triangles (the regions near D,G,K).  Again, the eigenfunction theory for isosceles right-angled triangles is very explicit, but this time the eigenvalue is simple and perturbation theory should be relatively straightforward.
4. Nearly 30-60-90 triangles (the regions near C,E,G,I,L,M).  Again, we have an explicit simple eigenfunction in the 30-60-90 case and an analysis should not be too difficult.

There are a number of stretching techniques (such as in the Laugesen-Siudeja paper) which are good for controlling how eigenvalues deform with respect to perturbations, and this may allow us to rigorously establish the first part of the hot spots conjecture, at least, for larger portions of the parameter space.

As for numerical verification of the second part of the conjecture, it appears that we have good finite element methods that seem to give accurate results in practice, but it remains to find a way to generate rigorous guarantees of accuracy and stability with respect to perturbations.  It may be best to focus on the super-equilateral acute isosceles case first, as there is now only one degree of freedom in the parameter space (the apex angle, which can vary between 60 and 90 degrees) and also a known anti-symmetry in the eigenfunction, both of which should cut down on the numerical work required.

I may have missed some other points in the above summary; please feel free to add your own summaries or other discussion below.