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Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:

(Faber-Krahn) Let $c$ be a positive number and $B$ the ball of volume $c$. Then $$\lambda_1(B) = \min\{\lambda_1(\Omega), \Omega\ \text{open subset of}\ \mathbb{R}^n, |\Omega| = c\}.$$

I am considering the question of minimizing $\lambda_1$ in the class of polygons with a given number $N$ as sides. If we denote by $\mathcal{P}_N$ the class of plane polygons with at most $N$ edges, then it is known that the problem $$\min\{\lambda_1(\Omega), \Omega \in \mathcal{P}_N, |\Omega| = a\}$$ has a solution. This one has exactly $N$ edges. For the case $N=3$, it has been proven the equilateral triangle minimizes $\lambda_1$. For $N=4$, the square minimizes $\lambda_1$. (Both proof uses the properties of the Steiner symmetrization of $\Omega$. The original proof was due to Pólya. Unfortuantely I could not find the original paper, but the proof can also be found in Extremum problems for Eigenvalues of Elliptic Operators by Henrot.)

Question: Is there a general result that the regular $N$-gon have the least first eigenvalue among all the $N$-gons of given area for $N \geq 5$?

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  • $\begingroup$ Can you include, in your question, the literature for N = 3,4? $\endgroup$ – Willie Wong Jun 4 '15 at 7:47
  • $\begingroup$ I edited the question :) $\endgroup$ – GavinZZZ Jun 4 '15 at 8:07
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    $\begingroup$ This is well known to be open. I think that the $N=3,4$ cases can be found in Polya and Szego's book "Isoperimetric Inequalities in Mathematical Physics." But the proof is not hard if you know that Steiner symmetrization decreases the first eigenvalue. $\endgroup$ – Otis Chodosh Jun 13 '15 at 17:51
  • $\begingroup$ @WillieWong The case $n=3,4$ was proven by Polya. See, for instance Henrot's paper. As far as I understand, for $n\geq 5$, Steiner symmetrization method increases the number of sides. $\endgroup$ – BigM Mar 2 '16 at 19:56
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For $N \geq 5$ it is still not known if the $N$-gon which minimizes the first eigenvalue under area constraint (which exists), is the regular one. I have done some numerical computations which suggest that the regular polygons are indeed optimal. You can see the numerical ideas here (recent version) or here (old version).

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So few people are working on this problem. I have noticed that results are scattered far and wide. So, I am calculating and compiling data related to the eigenvalues of the Laplacian within polygons. For example, below is a list of the principal Dirichlet eigenvalues within regular polygons (with area Pi, not inscribe in a unit-radius circle), all correctly rounded to 27 decimal places. I actually bounded them to within a relative error of at most 1E-30; and the pentagon, I have to 1E-500. But, below is a good list. The first two are known in closed form, and the last entry is the square of the first root of the Bessel function J_0(x)=0. This last one is the number the sequence is approaching. One can use the formula L = j01^2*(1+4*zeta(3)/N^3+ O(1/N^5)) to estimate the eigenvalue, with improving results as N (the number of polygon sides) increases. I do not use that formula, and I challenge anyone to figure out the non-zero fifth order term. (Incidentally, each eigenvalue shown from 127 sides and up takes about a day of CPU time. I'm doing the 256 sided polygon now, but I can't get thirty digits. I'm at only about 20 digits now, 5.7831876203689428759... where the trailing digits are yet uncertain.) -Bob Jones


3 7.255197456936871402376313031 <--- 4*Pi/sqrt(3)
4 6.283185307179586476925286767 <--- 2*Pi
5 6.022137932042633878298008710
6 5.917417831613661215688574577
7 5.866449312655985857712474942
8 5.838491433592442850516640380
9 5.821826802270265731735546444
10 5.811260359219116022788816469
11 5.804230636717400721878394453
12 5.799369804356500079315025311
13 5.795900266856014709790771063
14 5.793357005271194553273227079
15 5.791450010651579975693848498
16 5.789991899990208534349752214
17 5.788857871981104698617196635
18 5.787962591857846864212568380
19 5.787246351381961243008036645
20 5.786666514140372213530912962
21 5.786192077596844273028203757
22 5.785800129428365027574586044
23 5.785473486454901632048264070
24 5.785199089790024091834463613
25 5.784966894130423501418670684
26 5.784769086314842977992274718
27 5.784599527236484640593222827
28 5.784453347751719951196794842
29 5.784326652365411207380293386
30 5.784216299392264044119036734
31 5.784119736080032703344528385
32 5.784034873702444318330507487
33 5.783959992040508812335032523
34 5.783893665694809252033476569
35 5.783834706770988202840700005
36 5.783782119955880627699919966
37 5.783735067049846291962440637
38 5.783692838773292267706517922
39 5.783654832210871143386207911
40 5.783620532655973576951368559
41 5.783589498912728857243541088
42 5.783561351331960679963744950
43 5.783535762021971971277446762
44 5.783512446799268033474803358
45 5.783491158538856302913858879
46 5.783471681656168110105137225
47 5.783453827508463297379645914
48 5.783437430546865419250636065
49 5.783422345083940177286945517
50 5.783408442568212994780116196
51 5.783395609277903442166398140
52 5.783383744362702700019559015
53 5.783372758175598244048049661
54 5.783362570847292742897314144
55 5.783353111064236385644810842
56 5.783344315018129354638447816
57 5.783336125500292103090369355
58 5.783328491118809153913672275
59 5.783321365620033853737939611
60 5.783314707299059428210797800
61 5.783308478486244250571058736
62 5.783302645098928410170380598
63 5.783297176249175699422050587
64 5.783292043899785027461125829
65 5.783287222561990216101379319
66 5.783282689029249211147849046
67 5.783278422142346980297970170
68 5.783274402581728387648667337
69 5.783270612683560625466506793
70 5.783267036276517720104953505
71 5.783263658536697267505357057
72 5.783260465858434270390794564
73 5.783257445739078946040870361
74 5.783254586676063084321584559
75 5.783251878074799951907284413
76 5.783249310166151671634629193
77 5.783246873932360298530618523
78 5.783244561040478512305563119
79 5.783242363782456338656120879
80 5.783240275021144443460263290
81 5.783238288141564708788818914
82 5.783236397006877016631166830
83 5.783234595918539141935169159
84 5.783232879580215836619574007
85 5.783231243065044797761869160
86 5.783229681785912299987294190
87 5.783228191468430723800997531
127 5.783199538123680412174552014
128 5.783199222432098956985238320
129 5.783198916453726829015452454
130 5.783198619817847494322697718
5.783185962946784521175995758 <-- j_{0,1}^2 (J_0(j_{0,1})=0)

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    $\begingroup$ The subject per se might be useful, but this is probably not the right place to "publish" this result. $\endgroup$ – Alex Degtyarev Jun 24 '15 at 9:50
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    $\begingroup$ Thanks, but I read this:"Thanks for contributing an answer to MathOverflow! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience...." and thought I was being helpful and relevant about the question. --Bob $\endgroup$ – user122986 Jun 25 '15 at 11:48
  • $\begingroup$ I believe this is an update to the answer of @user122986: arxiv.org/pdf/1602.08636v1.pdf $\endgroup$ – Neal Oct 16 '16 at 0:00

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