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Steklov eigenvalues of the classical planar domains
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Numbers
$\Omega$
$n$ 
$\sigma_n(\Omega)$
unit square
1:
0
comment: The constant eigenfunction gives the simple zero eigenvalue.
equals: T485#unit-square,1
unit square
2:
1.376505484672534777969704669646540416306884308102950132316368378803244737482743823273280356666625695
unit square
3:
1.376505484672534777969704669646540416306884308102950132316368378803244737482743823273280356666625695
equals: unit-square,2
unit square
4:
2
comment: The bilinear eigenfunction $(x-1/2)(y-1/2)$ gives the simple eigenvalue $2$.
unit square
5:
4.647275506863445371216208003531306685799308298244743833420577757694473677274939995073985922228099628
unit square
6:
4.647275506863445371216208003531306685799308298244743833420577757694473677274939995073985922228099628
equals: unit-square,5
unit square
7:
4.780778410211634021273076773306499429048230092470448054516257288624903684839760023781799625932123567
unit square
8:
4.780778410211634021273076773306499429048230092470448054516257288624903684839760023781799625932123567
equals: unit-square,7
unit square
9:
7.848666046489503231133741232205791820804059917593312645848896095257707495215286706950400282440817556
unit square
10:
7.848666046489503231133741232205791820804059917593312645848896095257707495215286706950400282440817556
equals: unit-square,9
unit square
11:
7.859309013560367598717155663171781527914055355563192586572347266617433847801696937652254395253281117
unit square
12:
7.859309013560367598717155663171781527914055355563192586572347266617433847801696937652254395253281117
equals: unit-square,11
unit square
13:
10.99523893673765154359255492941557138287369488636514208641616810580878986882924558858993792472015157
unit square
14:
10.99523893673765154359255492941557138287369488636514208641616810580878986882924558858993792472015157
equals: unit-square,13
unit square
15:
10.99590967102146919282160729382434231206179620603038809214925773470335212312576603870138529381736182
unit square
16:
10.99590967102146919282160729382434231206179620603038809214925773470335212312576603870138529381736182
equals: unit-square,15
unit square
17:
14.13714789368830811280553053743736449047564191965488422102927052818591959609330857522248028236973223
unit square
18:
14.13714789368830811280553053743736449047564191965488422102927052818591959609330857522248028236973223
equals: unit-square,17
unit square
19:
14.13718598870057946262521727787632762769685786799312000120663775605164793626317343609883477394311629
unit square
20:
14.13718598870057946262521727787632762769685786799312000120663775605164793626317343609883477394311629
equals: unit-square,19
$1\times2$ rectangle
1:
0
comment: The constant eigenfunction gives the simple zero eigenvalue.
equals: T485#rectangle-1x2,1
$1\times2$ rectangle
2:
0.5553099889581934229290951374879191546833831117399663707065640585347479721806833751631809989354764857
$1\times2$ rectangle
3:
1.527108146110069486890969519550984128448366365302832606944933685577176834777860492913222189545249817
$1\times2$ rectangle
4:
1.629408877981245049394414348883020380950810721120561268987023674408076976825081124628150496627214058
$1\times2$ rectangle
5:
2.052218364421001384321402499065791967393116734067921697680858646174641278252862621548458508685805889
$1\times2$ rectangle
6:
2.781026006806295297580781282788570781869224407357937590669295022562000841269168067791553900864491146
$1\times2$ rectangle
7:
3.796674745730967141135615140302166703843507322639442164942556121847498542095845958785858834214576961
$1\times2$ rectangle
8:
4.067192909762319958411073791939920242080189285298037117785623459469011012086829017613390994224045970
$1\times2$ rectangle
9:
4.711790033006847584594208658748179652461458770170762029297121861482876102348772626051564484044803103
$1\times2$ rectangle
10:
4.712988385245167448394340357273525306690979917241226978069827300831505326057782493148520303059298094
$1\times2$ rectangle
11:
5.457162732839665912178660487534282369143701904667670575453701309033161745219546377977050571294906726
$1\times2$ rectangle
12:
5.539048792019246724793671047003563800975856095951471934731755129073754339764428005152326616865015386
$1\times2$ rectangle
13:
7.057416270710266319246843780641928232940153051826529100556146166095911947865185159208647228696522840
$1\times2$ rectangle
14:
7.079787315697601945316780684407149250426622526432389880071065635319673657567695859868002912190183565
$1\times2$ rectangle
15:
7.853979568162248759993713485815119912325422496005655113039055060329596566415694118063132633186286256
$1\times2$ rectangle
16:
7.853983699789739805279012434331496268718449062915987938441862065504199379820801788276201994300126425
$1\times2$ rectangle
17:
8.636499499824641907023510004027014710334624842846060165833814261314982562383194506030407664174247883
$1\times2$ rectangle
18:
8.642262072175549676594132889830317417271993919533296144104294277167179177266138876594244505158300743
$1\times2$ rectangle
19:
10.20946162320401778281940999455187214556666300378715461066036037359817960658644790223972804786576785
$1\times2$ rectangle
20:
10.21089072758693944501142516487830184306828558278254950797260838310928266976580144482800482531742520
unit disk
1:
0
comment: The constant eigenfunction gives the simple zero eigenvalue.
equals: T485#unit-disk,1
unit disk
2:
1
unit disk
3:
1
equals: unit-disk,2
unit disk
4:
2
unit disk
5:
2
equals: unit-disk,4
unit disk
6:
3
unit disk
7:
3
equals: unit-disk,6
unit disk
8:
4
unit disk
9:
4
equals: unit-disk,8
unit disk
10:
5
unit disk
11:
5
equals: unit-disk,10
unit disk
12:
6
unit disk
13:
6
equals: unit-disk,12
unit disk
14:
7
unit disk
15:
7
equals: unit-disk,14
unit disk
16:
8
unit disk
17:
8
equals: unit-disk,16
unit disk
18:
9
unit disk
19:
9
equals: unit-disk,18
unit disk
20:
10
annulus of radii $1$ and $2$
1:
0
comment: The constant eigenfunction gives the simple zero eigenvalue.
equals: unit-square,1
annulus of radii $1$ and $2$
2:
0.2192235935955848625446475360064807437132001936565948914003416067262614134155946016030340872973289258
equals: Algebraic_numbers_of_degree_2#2,-5,1,1
comment: The first positive eigenvalue is $(5-\sqrt{17})/4$.
annulus of radii $1$ and $2$
3:
0.2192235935955848625446475360064807437132001936565948914003416067262614134155946016030340872973289258
equals: Algebraic_numbers_of_degree_2#2,-5,1,1
annulus of radii $1$ and $2$
4:
0.7566018867943396188679339622377359283016377366584878679337018551019977090414881941458973013109615895
annulus of radii $1$ and $2$
5:
0.7566018867943396188679339622377359283016377366584878679337018551019977090414881941458973013109615895
equals: annulus-1-2,4
annulus of radii $1$ and $2$
6:
1.378544372682565313710678542198510805348788754379912684293979675263898535274481544124362341171573070
annulus of radii $1$ and $2$
7:
1.378544372682565313710678542198510805348788754379912684293979675263898535274481544124362341171573070
equals: annulus-1-2,6
annulus of radii $1$ and $2$
8:
1.955011698384984519437999104810827841999270893921281608079942085457196730660238335055184193442199475
annulus of radii $1$ and $2$
9:
1.955011698384984519437999104810827841999270893921281608079942085457196730660238335055184193442199475
equals: annulus-1-2,8
annulus of radii $1$ and $2$
10:
2.164042561333445111039887021502838206139968931229478901203174110396663828771777619828289934340259517
annulus of radii $1$ and $2$
11:
2.280776406404415137455352463993519256286799806343405108599658393273738586584405398396965912702671074
equals: Algebraic_numbers_of_degree_2#2,-5,1,2
annulus of radii $1$ and $2$
12:
2.280776406404415137455352463993519256286799806343405108599658393273738586584405398396965912702671074
equals: Algebraic_numbers_of_degree_2#2,-5,1,2
annulus of radii $1$ and $2$
13:
2.485506277764000834008624838171747187649811583521376618674480195607436505517051797216523355003796730
annulus of radii $1$ and $2$
14:
2.485506277764000834008624838171747187649811583521376618674480195607436505517051797216523355003796730
equals: annulus-1-2,13
annulus of radii $1$ and $2$
15:
2.643398113205660381132066037762264071698362263341512132066298144898002290958511805854102698689038410
annulus of radii $1$ and $2$
16:
2.643398113205660381132066037762264071698362263341512132066298144898002290958511805854102698689038410
equals: annulus-1-2,15
annulus of radii $1$ and $2$
17:
2.995617220179462083423092622243912159937182820029044333567620358101559746632484389199059965598288465
annulus of radii $1$ and $2$
18:
2.995617220179462083423092622243912159937182820029044333567620358101559746632484389199059965598288465
equals: annulus-1-2,17
annulus of radii $1$ and $2$
19:
3.264312770174577543432178600658632051794068388477230172848877467593244321868375598732780515971284073
annulus of radii $1$ and $2$
20:
3.264312770174577543432178600658632051794068388477230172848877467593244321868375598732780515971284073
equals: annulus-1-2,19
Definition
The Steklov eigenvalue $\sigma_n(\Omega)$ is an eigenvalue of $\Delta u=0$ in the planar domain $\Omega$, with $\partial_\nu u=\sigma u$ on its entire boundary and $\nu$ the outward unit normal [3].
Parameters
$\Omega$
—   domain ($\Omega$ is one of the listed classical planar domains.)
$n$
—   eigenvalue index ($n\geq1$)
Formulas
(1)
For the unit disk, $\sigma_1=0$ and $\sigma_{2m}=\sigma_{2m+1}=m$ for every integer $m\geq1$. The nonconstant eigenfunctions are $r^m\cos(m\theta)$ and $r^m\sin(m\theta)$ [1], Example 1.3.1.
(2)
For $(-a,a)\times(-b,b)$, take both assignments $(p,q)=(a,b)$ and $(b,a)$. Define $T_p(\alpha)=\alpha\cot(p\alpha)$ for a sine factor or $T_p(\alpha)=-\alpha\tan(p\alpha)$ for a cosine factor, and $H_q(\alpha)=\alpha\tanh(q\alpha)$ for a hyperbolic cosine factor or $H_q(\alpha)=\alpha\coth(q\alpha)$ for a hyperbolic sine factor. Each choice and positive solution of $T_p(\alpha)=H_q(\alpha)$ contributes the eigenvalue $\sigma=H_q(\alpha)$. Together with $0$, these give the spectrum; when $a=b$, add the simple eigenvalue $1/a$ of the bilinear function $xy$. The two coordinate assignments contribute separate eigenfunctions even when $a=b$ [2], Lemma 2.4.
(3)
For $1<r<2$, the simple radial eigenvalues are $0$ and $3/(2\log2)$. For each integer $m\geq1$, put $C_m=\frac{3m}{2}\frac{4^m+1}{4^m-1}$. The two roots of $\sigma^2-C_m\sigma+m^2/2=0$ each contribute twice to the spectrum. Sort this multiset, including the radial eigenvalues, to obtain $\sigma_n$ [1], Example 4.2.5.
(4)
For the unit square, the separation parameter $\alpha$ in (2) satisfies $\cos\alpha\cosh\alpha=1$ for the cosine/hyperbolic-cosine and sine/hyperbolic-sine pairs, and $\cos\alpha\cosh\alpha=-1$ for the other two pairs. These are the clamped-clamped and clamped-free equations in the Euler-Bernoulli beam table.
(5)
For a dilation factor $c>0$, $\sigma_n(c\Omega)=\sigma_n(\Omega)/c$ [1], equation (4.1.1).
Comments
(6)
The domains have the sizes used in the Dirichlet table: the unit square is $(0,1)^2$, the rectangle is $(0,1)\times(0,2)$, the disk has radius $1$, and the annulus is $1<r<2$. No area or perimeter normalization is applied.
(7)
Boundary density is $1$. Eigenvalues are ordered nondecreasingly, with the simple zero eigenvalue at $n=1$, and repeated according to multiplicity. The disk and annulus have the two angular eigenfunctions $\cos(m\theta)$ and $\sin(m\theta)$ for each positive integer $m$. The stored quantity is $\sigma$, without multiplying by boundary length or taking a square root.
(8)
On the annulus, the boundary conditions are $-\partial_r u(1,\theta)=\sigma u(1,\theta)$ and $\partial_r u(2,\theta)=\sigma u(2,\theta)$. The two boundary components carry the same unit boundary density.
(9)
Translating a rectangle to its center leaves its spectrum unchanged. The half-widths are $a=b=1/2$ for the unit square and $(a,b)=(1/2,1)$ for the $1\times2$ rectangle. Completeness of the separated eigenfunctions follows from the Robin-product argument in [2], Lemma 2.1.
(10)
The domain $\Omega$ is an argument of the Steklov spectrum. Each row gives the same quantity, $\sigma_n(\Omega)$, on a specified shape with the sizes fixed in (6).
Programs
(P1)
Python (mpmath)
# Unit square: the first positive eigenvalue (indices n=2 and n=3).
import mpmath as mp
mp.mp.dps = 110
alpha = mp.findroot(lambda x: mp.cos(x)*mp.cosh(x)+1, (1, 2))
sigma_2 = alpha*mp.tanh(alpha/2)
print(mp.nstr(sigma_2, 100))
# For certified enclosures and all four domains, use the attached generate.py.
References
[1]
Alexandre Girouard and Iosif Polterovich, Spectral geometry of the Steklov problem (Survey article), Journal of Spectral Theory 7 (2017), 321–359. (arXiv) (doi)
[2]
Alexandre Girouard, Jean Lagacé, Iosif Polterovich and Alessandro Savo, The Steklov spectrum of cuboids, Mathematika 65 (2019), 272–310. (arXiv) (doi)
Links
Similar tables
Dirichlet_eigenvalues_of_the_classical_planar_domains —   the same domain geometries with a Dirichlet boundary condition
Neumann eigenvalues of the classical planar domains —   the same domain geometries with a Neumann boundary condition
Roots of the frequency equations of the Euler-Bernoulli beam —   its clamped-free and clamped-clamped roots also determine the square separation parameters
Zeros of cross-products of Bessel functions —   scalar characteristic roots for Dirichlet and Neumann annulus problems
Eigenvalues of the clamped plate —   fourth-order spectra of planar domains with clamped boundary conditions
Algebraic_numbers_of_degree_2 —   the two annulus eigenvalues of angular order one are roots of $2x^2-5x+1$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds modes $1\leq n\leq20$ for the unit square, $1\times2$ rectangle, unit disk and annulus of radii $1$ and $2$, including zero and counting multiplicity. These low modes give a compact reference for classical Steklov problems. The range ends at index $20$ even if a repeated eigenvalue continues at index $21$. The circular sectors, triangles, L-shape, Gordon-Webb-Wolpert drums and regular polygons in the Dirichlet table are omitted)
Relative precision: 100 significant decimal digits for approximate entries
How they were obtained:

The zero modes, all disk modes and the square bilinear mode are exact. Other entries are enclosed using Arb ball arithmetic and written to 100 significant digits. Rectangle roots are bracketed with certified opposite signs on branches where the trigonometric Robin quotient decreases and the hyperbolic quotient increases. The first sine/hyperbolic-sine branch has a positive root only when its hyperbolic half-width exceeds its trigonometric half-width; the square limit is the separately counted bilinear mode. Bounds on omitted branches and angular orders certify that the first 20 eigenvalues are included, and disjoint enclosures certify their order. The annulus formulas use exact rational coefficients, a ball square root and a ball logarithm. With 100 guard bits, the smallest measured relative precision exceeds 110 decimal digits.

more

Independent checks recompute the rectangle spectra at 140 and 180 decimal digits using mpmath, compare their energy integrals with their boundary mass, and compute the annulus modes from a symmetric Dirichlet-to-Neumann matrix. Controls recover the disk value 1 and the square bilinear value 2. The annulus boundary equations are checked with the negative radial derivative on the inner circle. All 80 written intervals are tested against the independent computations. The 18 nonpolynomial square modes also have their separation parameters compared with the stored beam roots in T486. The attached checks can repeat these tests on values read back from this draft.



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