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=== General domains ===
=== General domains ===


A counterexample to the hot spots conjecture for a domain with two holes was established in [BW1999].
A counterexample to the hot spots conjecture for a domain with two holes was established in [BW1999]. On the other hand, the conjecture was established for convex domains with two axes of symmetry in [JN2000].


== Possible counterexamples ==
== Possible counterexamples ==

Revision as of 10:06, 9 June 2012

The hotspots conjecture can be expressed in simple English as:

Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.

In mathematical terms, we consider a two-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We then conjecture that

For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D

This conjecture has been proven for some domains and proven to be false for others. In particular it has been proven to be true for obtuse and right triangles, but the case of an acute triangle remains open. The proposal is that we prove the Hot Spots conjecture for acute triangles!

Note: strictly speaking, the conjecture is only believed to hold for generic solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic eigenvectors of the second eigenvalue of the Laplacian attain their maximum on the boundary.

Threads

Possible approaches

Combinatorial approach

Sturm comparison approach

How about approximating the eigenfunction by polynomials? The Neumann boundary conditions already imply that the corners are critical points. Perhaps starting with that observation, for sufficiently low degree, the geometry of the triangle implies that one of them must be a max/min. In an ideal world, a 2d version of the Sturm comparison theorem (if one exists) could then show that this feature of the polynomial approximation remains true for the actual eigenfunction.

In my understanding, Sturm-type results allow you to establish some qualitative property for solutions of an ODE (a 1d elliptic problem) provided that the desired property holds for a nearby solution of a nearby ODE. The property usually considered is the presence of a zero (node) of an eigenfunction in some interval. I think a similar result holds for critical points of eigenfunctions.

Say you could establish that the second eigenfunction of the Laplacian on a right angled triangle satisfies the desired maximum principle by showing that its only critical points are at the vertices (so that whatever the maximum is, it would have to be at one of these points). Then appealing to a comparison theorem might be able to show that the same property holds for almost-right acute triangles. Mind you, this is all speculation at this point…

Ah, so maybe a Sturm-type result can make rigorous the statement: “the second eigenfunction depends continuously on the domain”? By physical considerations, this statement ought to be true.

But then it seems this would only work to prove the conjecture for triangles close to right triangles?

As a side note, if such a continuity statement were to hold then the hot spots conjecture for acute triangles must be true by the following *non-math* proof: If it weren’t true then by continuity there would be an open set of angles where it failed to hold. But after simulating many triplets of angles the conjecture always holds. C’mon, What are the odds we missed that open set? :-)

It appears that Sturm’s classical work has far reaching generalizations, as described for instance in this monograph: Kurt Kreith, Oscillation Theory LNM-324, (Springer, 1973). In particular, Chapter 3 features some comparison theorems for solutions to elliptic equations.

Subdivision

Maybe ideas based on self-similarity of the whole triangle to its 4 pieces can help (i.e. modeling the whole triangle as its scaled copies + the heat contact). Then without going into the graph approximation (which looks fruitful anyway), one can see some properties.

One advantage in the graph case is that after dividing and dividing the triangles you get to the graph G_1 which is simply a tree of four nodes and there I think the theorem shouldn’t be too hard to prove. And from there there might be an argument by induction. In the continuous case, no matter how many times you subdivide the triangle, after zooming it it is still the same triangle.

On the other hand, in the continuous case, each of the sub-triangles is truly the same as the larger triangle. Does anyone know of good examples where self-similarity techniques are used to solve a problem?

In either case, there is the issue that while the biggest triangle has Neumann boundaries, the interior subtriangles have non-Neumann boundaries…

Something that is true and uses that the whole triangle is the union of the 4 congruent pieces is the following. From each eigenfunction of the triangle with eigenvalue lambda we can build another eigenfunction with eigenvalue 4 lambda by scaling the triangle to half length, and then using even reflection through the edges. Note that this eigenfunction will have an interior maximum since every point on the boundary has a corresponding point inside (on the boundary of the triangle in the middle). I’m skeptical this observation could be of any use.

Hmm… but, apart from the case of an equilateral triangle, when you divide a triangle into four sub-triangles, I don’t believe that the sub-triangles are reflections of one another.

Brownian motion

Suppose I start off a Brownian motion at time 0 at some point [math]\displaystyle{ x }[/math] in the triangle, and the Brownian motion reflects off the sides of the triangles. At time $latex t$ the probability density for the Brownian motion to be at location [math]\displaystyle{ y }[/math] is given by the solution [math]\displaystyle{ v(t,y) }[/math] of the heat equation with [math]\displaystyle{ v(0,y)=\delta(x-y) }[/math] and Neumann boundary conditions. So if I can show that the Brownian motion is eventually more likely to be in a corner (per unit area) than anywhere else, that will gives us the result.

Here's a way to approach it: Suppose I fix [math]\displaystyle{ x }[/math]. I ask how many ways are there for Brownian motion to get from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math] in time [math]\displaystyle{ t }[/math]. For [math]\displaystyle{ t }[/math] large, If [math]\displaystyle{ y }[/math] is in a corner I might imagine that there are more ways of Brownian motion to get to [math]\displaystyle{ y }[/math] than otherwise, because the Brownian motion has all the ways it would in free space, but also a ton of new ways that involve bouncing off the walls. Perhaps you can using some argument showing that there are more paths into some corners than anywhere else in the triangle. (Perhaps, using the Strong Markov Property and Coupling?)

This approach is also discussed at http://www.math.missouri.edu/~evanslc/Polymath/Polymath.pdf, and a similar approach was pursued in [BB1999].

Special cases

Isosceles triangles

Say the triangle has vertices [math]\displaystyle{ (a,0) }[/math], [math]\displaystyle{ (-a,0) }[/math] and [math]\displaystyle{ (0,h) }[/math]. If [math]\displaystyle{ u(x,y) }[/math] is the second eigenfunction, then also is [math]\displaystyle{ u(-x,y) }[/math] and their sum [math]\displaystyle{ v(x,y)=u(x,y)+u(-x,y) }[/math]. The function v is even in x, so it is also a Neumann eigenvalue of the right triangle with vertices [math]\displaystyle{ (0,0) }[/math], [math]\displaystyle{ (0,h) }[/math] and [math]\displaystyle{ (a,0) }[/math]. According to the description of the problem the case of the right triangle has already been worked out, so this would reduce to that case unless v is identically zero.

If v is identically zero, it means that the second eigenfunction u was odd in x. In particular [math]\displaystyle{ u(0,y)=0 }[/math] for all y. The function u would be the first eigenfunction on the same right triangle as above but now with Dirichlet boundary condition on x=0 and Neumann in the other two edges. This function cannot change sign (otherwise |v| has less energy). I would expect its maximum to take place at (a,0) but I don't know how to prove it (maybe some rearrangement along lines?).

But how do you know that it is still the second eigenfunction(as opposed to some other eigen function)?


Equilateral triangles

Quoting from [McC2011]: "in 1833, Gabriel Lam´e discovered analytical formulae for the complete eigenstructure of the Laplacian on the equilateral triangle under either Dirichlet or Neumann boundary conditions and a portion of the corresponding eigenstructure under a Robin boundary condition. Surprisingly, the associated eigenfunctions are also trigonometric. The physical context for his pioneering investigation was the propagation of heat throughout polyhedral bodies."

[McC2002] gives a complete description of Lame’s eigenfunctions for an equilateral triangle with a Neumann boundary coundition (del U)/(del n) = 0 , n the normal to the triangular boundary, U an eigenfuntion. In section 8 on Modal Properties, he gives beautiful expressions for the eigenfunctions in terms of pairs of integers (m, n) in Equations 8.1 and 8.2, where (8.1) covers what McCartin calls the symmetric, and (8.2) the antisymmetric modes (respectively).

I’m intrigued by whether the modes in equations (8.1) and (8.2) always attain their max and min on the boundary of the triangular region (equilateral triangle); the 3D plots of several modes by McCartin seems consistent with max/min always being attained on the boundary; Lame solved the equilateral triangle case, in the sense that he gave a complete set of eigenfunctions for the Laplacian eigenvalue problem with the (del U)/(del n) = 0 boundary value condition.

The second eigenvalue must, by a symmetry argument, have multiplicity at least two. The refined conjecture calls for a simple eigenvalue in the scalene case, and eigenfunction extremes at the two smaller angles. Has anyone done numerical work yet to estimate how sensitive the eigenvalue degeneracy lifting is to perturbations of a starting equilateral triangle? Any proof will have to address the issue that the third eigenvalue may be arbitrarily close to the second and has an eigenfunction with extremes at a different pair of corners.

Here is some preliminary data. As I am not sure how to simulate the true eigenfunctions of the triangle, the following is for the graph whose eigenvectors should roughly approximate the true eigenfunctions:

For the case that a=b=c=1 (Equilateral Triangle): HotSpotsAny(64,1,1,1,2) yields the corresponding eigenvalue -0.001070825047269 HotSpotsAny(64,1,1,1,3) yields the corresponding eigenvalue -0.001070825047269 HotSpotsAny(64,1,1,1,4) yields the corresponding eigenvalue -0.003211901603854

We see that indeed the eigenvalue -0.001070825047269 has multiplicity two.

Perturbing a slightly we have that for a=1.1, b = 1, c = 1 (Isosceles Triangle where the odd angle is larger than the other two): HotSpotsAny(64,1.1,1,1,2) yields the corresponding eigenvalue -0.001078552707489 HotSpotsAny(64,1.1,1,1,3) yields the corresponding eigenvalue -0.001131412869938

Whereas for a=.9, b = 1, c = 1 (Isosceles Triangle where the odd angle is smaller than the other two): HotSpotsAny(64,.9,1,1,2) yields the corresponding eigenvalue -0.001004876221957 HotSpotsAny(64,.9,1,1,3) yields the corresponding eigenvalue -0.001062028119964

In either case we see that the third eigenvalue is perturbed away from the second.

What strikes me as interesting is how different the outcome is in increasing a by 0.1 versus decreasing a by 0.1. In the former case, the second eigenvalue barely changes whereas in the latter case the second eigenvalue changes quite a bit. I imagine this ties into the heuristic “sharp corners insulate heat” — reducing a to .9 produces a sharper corner leading to more heat insulation and a much smaller (in absolute value) second eigenvalue, whereas increasing a to 1.1 makes that corner less sharp but the other two corners are still relatively sharp so the heat insulation isn’t affected as much. Just a guess though…

It looks like the degeneracy lifting might be linear. It’s the ratio of the second and third eigenvalues that matters more, which is close to the same in either case, and approximately half as far from 1 as the perturbation. Using (9/10, 1, 1) should be the same as using (1, 10/9, 10/9) and then scaling all eigenvalues by 90%.

Obtuse triangles

The obtuse case was settled in [BB1999] using a reflected Brownian motion approach. The “hot” and “cold” spots are located at the most distant vertices.

Right-angled triangles

The right-angled case of the conjecture is known (what is the reference?).

General domains

A counterexample to the hot spots conjecture for a domain with two holes was established in [BW1999]. On the other hand, the conjecture was established for convex domains with two axes of symmetry in [JN2000].

Possible counterexamples

If we could find an heat distribution on a right triangle such that the hot spot is never on the vertices or the hypotenuse or the short side then could we combine two such heat distributions symmetrically in an isosceles acute triangle and get an isosceles acute triangle such that the hot spots are always on the interior?

An issue is that if you reflect a right triangle and its eigenfunction the eigenfunction you get for the larger triangle might not be the *second* eigenfunction.

Miscellaneous remarks

If the sides of the triangle have lengths A, B, and C, currently you are using edge weights a=A, b=B, and c=C. This solves a different heat equation, when you pass to the limit, than the one you want. The assumption that each small triangle is at a uniform temperature gives an extra boost to the overall heat conduction rate in those directions along which the small triangles are longer, so in the limit you are modeling heat conduction in a medium with anisotropic thermal properties. It is just as though you started with a material of extremely high thermal conductivity and then sliced it in three different directions, with three different spacings, to insert thin strips of insulating material of a constant thickness.

Based on some edge cases, I suspect that the correct formulas for edge weights to model an isotropic material in the limit are as follows: [math]\displaystyle{ a = 1/(-A^2+B^2+C^2) }[/math], [math]\displaystyle{ b=1/(A^2-B^2+C^2) }[/math], [math]\displaystyle{ c=1/(A^2+B^2-C^2) }[/math]. Interestingly enough, these formulas are only defined (with positive weights) in the case of acute triangles, which suggests that this approach, if it works, may not provide an independent proof of the known cases.


Software

  • A MATLAB program used to generate and display the eigenvectors of the graphs G_n. The input format is HotSpotsAny(n,a,b,c,e), where a,b,c are the edge weights, n is the number of rows in the graph, and e determines which eigenvector of the graph Laplacian is displayed. (Note that in the proposal write up the graph G_n has 2^n rows so if you wanted to simulate the Fiedler vector for G_6 with edge weights a=1,b=2,c=3 then you would type HotSpotsAny(64,1,2,3,2)). For large n, the eigenvectors of the graph Laplacian should approximate the eigenfunctions of the Neumann Laplacian of the corresponding triangle… so the program can be used to roughly simulate the true eigenfunctions as well.

Numerics

Using a finite element method, here are the eigenfunctions of the Neumann Laplacian for a couple of acute triangles:

The domain is subdivided into a finite number N of smaller triangles. On each, I assumed the eigenfunction can be represented by a linear polynomial. Across interfaces, continuity is enforced. I used Matlab, and implemented a first order conforming finite element method. In other words, I obtained an approximation from a finite-dimensional subspace consisting of piece-wise linear polynomials. The approximation is found by considering the eigenvalue problem recast in variational form. This strategy reduces the eigenvalue question to that of finding the second eigenfunction of a finite-dimensional matrix, which in turn is done using an iterative method. As N becomes large, my approximations should converge to the true desired eigenfunction. This follows from standard arguments in numerical analysis.

Additionally, I've computed a couple of approximate solutions to the heat equation in acute triangles. The initial condition is chosen to have an interior 'bump', and I wanted to see where this bump moved. Again, I've plotted the contour lines of the solutions as well, and one can see the bump both smoothing out, and migrating to the sharper corners:

I think in the neighbourhood of the corner with interior angle [math]\displaystyle{ \frac{\pi}{\alpha} }[/math], the asymptotic behaviour of the (nonconstant) eigenfunctions should be of the form [math]\displaystyle{ r^\alpha \cos(\alpha \theta) + o(r^{\alpha}) }[/math], where r is the radial distance from the corner.

Next, if one had an unbounded wedge of interior angle [math]\displaystyle{ \frac{\pi}{\alpha} }[/math], the (non-constant) eigenfunctions of the Neumann Laplacian would be given in terms of [math]\displaystyle{ P_n(r,\theta) = J_{\alpha, n} (\sqrt(\lambda) r) cos(\alpha n \theta), n=0,1,2\ldots }[/math]. The [math]\displaystyle{ J_{\alpha,n} }[/math] are Bessel functions of the first kind. The spectrum for this problem is continuous. When we consider the Neumann problem on a bounded wedge, the spectrum becomes discrete.

This makes me think approximating the second eigenfunction of the Neumann Laplacian in terms of a linear combination of such [math]\displaystyle{ P_n(r,\theta) }[/math] could be illuminating.

This idea is not new, and a numerical strategy along these lines for the Dirichlet problem was described in [FHM1967]. Betcke and Trefethen have a nice recent paper [BT2005] on an improved variant of this 'method of particular solutions'.

Bibliography

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