I just started a new thread to help organise all the new discussion without being cluttered by the older comments.

]]>The analysis there allows us to use piecewise conforming P1 elements, and obtain _computable_ constants, such that one can obtain the a posteriori bounds . Here the computed eigenvalue is an approximation of the kth true eigenvalue on a polygonal domain.

I’m re-doing the computations of the first two eigenvalues using this strategy, to get verified and validated bounds on the spectral gap in various parts of angle parameter space. Thanks to Bartek’s argument, I think the region of parameter space to explore is nicely reduced.

]]>Very nice indeed! I will try to write a new blog post for this polymath project with a brief summary of some of the new ideas here (this thread is getting too long to follow easily). The reduction to eliminating critical points on edges rather than the interior looks particularly promising, and it is great that the narrow triangles have been eliminated as they were looking a bit difficult to treat from the numerical perspective.

I have some ideas about how to push further the analysis of the nodal domains of a derivative of u (used in your proof of Theorem 1.3). Firstly it seems that if one uses the Neumann boundary condition of u and some integration by parts, one can compute rather exactly the functional of various nodal components of this derivative. Also there appear to be quite a few nodal domains: if one takes an edge where u changes sign, and assumes that there is at least one critical point in that edge, it seems to me that there must in fact be two critical points on that edge (because u has a local maximum at one end of the edge and a local minimum at the other). I haven’t fleshed out the details of this yet though.

]]>Bartek, this is very nice work! I have not checked all the details but it looks promising indeed. It seems that you’ve got a very nice proof (Theorem 1.4) for the triangles which have one angle small (less than pi/6)…. and these are usually cases in which naive mesh refinements of FEM don’t do so well. Good work!

]]>There is certainly some room for improvement. In particular when the eigenvalue bound in Lemma 1.6 does not hold, one might try to use a more complicated eigenfunction instead of cosine. I have done some Mathematica/numerical experiments and for equilateral triangle one might take a rectangle eigenfunction and eliminate critical points on the middle 70% of each side. I wasn’t able to find any eigenfunction, that would give good results for critical points near vertices.

Also, if Lemma 1.6 could be somehow applied to all sides, then the positive minimum (negative maximum) case can happen only once, since nodal line cannot start and end on the same side. This would mean no critical points on two sides, hence hot-spots conjecture would hold.

]]>I am not sure, I will have to think about this. There are certain conditions that must be satisfied first. E.g. eigenvalue bound that guarantees that cosine eigenfunction covers the whole triangle (cosine decreases on the whole triangle giving negative outer normal derivative). Also, cosine would not be the eigenfunction for other elliptic operators, and monotonicity (negative normal derivative condition) would not be immediate.

But this sounds like a possible approach.

]]>Bartek, I’m trying to understand from Miyamoto’s paper if his argument on the isoceles triangle could be extended to a generic constant-coefficient elliptic operator. I don’t see an immediate obstacle, but is there?

In other words, suppose one could somehow construct a target eigenfunction (like his ) for the eigenvalue problem . Can one use a variant of his argument for the eigenvalue problem with this new operator, with boundary conditions ?

I ask because then one could map acute triangles to the reference triangle, and perhaps make the argument there on the perturbed Laplacian? His argument settles the conjecture on domains like the reference triangle.

]]>I agree that if one uses a non-radial test function then the condition on the normal derivative becomes messy to verify, but I think for a validated numeric approach it is still feasible to work out when the condition applies. One could imagine a brute force approach where one explores Miyamoto’s criterion for test solutions of the eigenfunction equation that consist of the zeroth and first angular modes (so something like a linear combination of and in polar coordinates around ). [EDIT: we need to work with instead to keep it a critical point at .] If one is lucky one might be able to cover the entire interior of the triangle just with these trial functions.

Incidentally, I was reading a 2004 paper of Filonov http://www.ams.org/journals/spmj/2005-16-02/S1061-0022-05-00857-5/S1061-0022-05-00857-5.pdf where he gives a short proof of Polya’s inequality that the second Neumann eigenvalue is dominated by the first Dirichlet eigenvalue in a Euclidean domain (actually he also proves the generalisation to higher eigenvalues by Friedlander) – this fact is used in Miyamoto’s argument to rule out loops in the zero set of z. It turns out that Filonov also uses an auxiliary solution to the eigenfunction equation, in this case the plane waves for some of magnitude . Indeed, if is a Dirichlet eigenfunction of eigenvalue , a short computation shows that any linear combination of and has Raleigh quotient equal to ; in particular one can take a linear combination of mean zero and conclude that is a lower bound for the Neumann eigenvalue (and with a bit more effort on can show that this is a strict inequality in two and higher dimensions, e.g. by varying ). It may be that a similar trick (i.e. working with instead of ) could boost the power of Miyamoto’s method (e.g. if we are somehow able to know on which sides of the triangle w-u is positive), but I haven’t played seriously with this yet.

]]>It would certainly be excellent (from the point of view of validated numerics) to be able to shrink the candidate search region for extrema as you describe, with computable quantities specifying where the search should focus.

If ellipses can be used instead, then presumably a local change to elliptical coordinates, followed by the use of Mathieu functions, will be useful.

For reference, here’s a nice article by Steve Zelditch in which he discusses the form (and properties) of eigenfunctions in some specific instances. http://www.ms.uky.edu/~perry/CBMS/EigenfsJune9.pdf

Also for reference (and possible future use) is this paper by Kuttler in which he summarized numerical approaches for some standard domains. http://epubs.siam.org/doi/abs/10.1137/1026033

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