TITLE: First eigenvalue of the Laplacian on a regular polygon
QUESTION [12 upvotes]: 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 $N \geq 5$?

REPLY [4 votes]: 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)