Elliptic Eigenvalue Problems

This page concerns elliptic eigenvalue problems of the general form \[ \mbox{ Find } (u,\lambda)\in\mathcal{H}\times\mathbb{C}\mbox{ such that }\int_\Omega A\nabla u\cdot\nabla v+(\mathbf{b}\cdot\nabla u+cu)v\,dx = \lambda\int_\Omega \beta uv\,dx\mbox{ for all }v\in \mathcal{H}~, \] where \(\mathcal{H}\) is some appropriate subspace of \(H^1(\Omega)\) which incorporates any (homogeneous) essential boundary conditions.

The Problems

\[\int_\Omega\nabla u\cdot\nabla v\,dx=\lambda\int_\Omega uv\,dx\]

Fix \(\alpha\in(1,2]\) and let \(\Omega\subset \mathbb{R}^2\) be the sector of the unit disk for which \( 0 < r < 1 \) and \( 0 < \theta < \alpha \pi \); when \(\alpha=2\), \(\Omega\) is the unit disk with a slit along the positive \(x\)-axis.

First-Kind Bessel Functions and Their Roots

\[ J_\sigma(z)=\left(\frac{z}{2}\right)^\sigma\sum_{k=0}^\infty\frac{(-1)^k}{k!\Gamma(k+\sigma+1)}\left(\frac{z}{2}\right)^{2k} \quad,\quad j_m(\sigma)=m^{th}\mbox{ positive root of }J_\sigma \]

Dirichlet condition on \(r=1\), Dirichlet conditions on \(\theta=0\) and \(\theta=\alpha\pi\)

All eigenpairs doubly-indexed, \( (u_{km},\lambda_{km})\): \[ u_{km}=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\sin(\sigma_k\,\theta)\quad,\quad \lambda_{km}=[j_m(\sigma_k)]^2\quad,\quad \sigma_k=\frac{k}{\alpha}\quad,\quad k\geq 1\;,\; m\geq 1 \]

Dirichlet condition on \(r=1\), Neumann conditions on \(\theta=0\) and \(\theta=\alpha\pi\)

All eigenpairs doubly-indexed, \( (u_{km},\lambda_{km})\): \[ u_{km}=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\cos(\sigma_k\,\theta)\quad,\quad \lambda_{km}=[j_m(\sigma_k)]^2\quad,\quad \sigma_k=\frac{k}{\alpha}\quad,\quad k\geq 0\;,\; m\geq 1 \]

Dirichlet condition on \(r=1\), Dirichlet condition on \(\theta=0\) and Neumann condition on \(\theta=\alpha\pi\)

All eigenpairs doubly-indexed, \( (u_{km},\lambda_{km})\): \[ u_{km}=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\sin(\sigma_k\,\theta)\quad,\quad \lambda_{km}=[j_m(\sigma_k)]^2\quad,\quad \sigma_k=\frac{2k+1}{2\alpha}\quad,\quad k\geq 0\;,\; m\geq 1 \]

Slit Disk (\(\alpha=2\)), Dirichlet/Neumann:

The first ten eigenvalues for \( 0 \leq k \leq 7 \), accurate to all 30 digits shown, up to rounding in the last digit.

k=0	k=1	k=2	k=3
7.73333653346596686390263803337	12.1871394680951290047505723560	17.3507761313694859586686502730	23.1993865387331719385298116070
34.8825215790904790430911907100	44.2575594035024472537244889622	54.3596015232094873965461826498	65.1767709083390084446143156735
81.7647023194374255502663810252	96.0716048388430901772947338831	111.109164189343629901279779194	126.870681508728170693543652387
148.385081445043562614168937452	167.625712420578303557539528801	187.598320803070079647608461010	208.298640528527733692491069300
234.744373345439788620322954495	258.919300357440362078091918285	283.826822825821542321768305561	309.463994348658829866563530533
340.842757625123711498282368313	369.952209262346068631425669745	399.794593580023538912908529450	430.367756267178609952304627478
466.680295946200469586902720763	500.724381475794183238478665262	535.501603402747510461337308750	571.010319011209343057658253475
612.257014004421679312402962975	651.235792102542719942360069398	690.947838917131069749831094445	731.391860906830073076873322232
777.572924032271491109121826445	821.486428982383575833591226923	866.133293385950192385198831099	911.512472473474305874440881571
962.628032449008542130660970732	1011.47628560801610204045214133	1061.05796312084933360076379021	1111.37220364314013210781706831
k=4	k=5	k=6	k=7
29.7145342842106938075714690649	36.8818928841180487579063517659	44.6899430974683906098000106225	53.1291788410994927833576705400
76.6978978017046035416048785377	88.9128264032049384243850896701	101.812416292492719069621675283	115.388452233110190920437947927
143.349064446158040032227622255	160.537281358588387414948116155	178.428565668674780381276777562	197.016500904227884270081263232
229.721824222112591812253100109	251.862795845034644081660610462	274.716449610594182764748668023	298.277762419288576970640911812
335.827306343422842911542053594	362.912943708380157494716304537	390.716946367958682711301651648	419.235312392571175123581640371
461.669044944096065292760123739	493.695495073021230243253067956	526.443956299934073827540705208	559.911179211391462545607569442
607.248454712575419803555986611	644.213643759938350432215465654	681.903324808450377564783571548	720.314813110147188859770072906
772.566193335015998808613121766	814.468904465270548113688850474	857.097873102817923700662127041	900.450846764162254443718796088
957.622600730245947047732883970	1004.46207234618030305792837226	1052.02910294995276439347169279	1100.32177730883324768231690129
1162.41786709784105416490040334	1214.19359794648075963125492979	1266.69787495878397894333570136	1319.92905059524769445013120138

Contour plots of the eigenfunctions corresponding to the smallest six eigenvalues (colored in red above). The first and sixth have the strongest singularity at the origin, \(r^{1/4}\).

Slit Disk (\(\alpha=2\)), Neumann/Neumann:

The first ten eigenvalues for \( 0 \leq k \leq 7 \), accurate to all 30 digits shown, up to rounding in the last digit.

k=0	k=1	k=2	k=3
5.78318596294678452117599575846	9.86960440108935861883449099988	14.6819706421238932572197777686	20.1907285564266299745230737654
30.4712623436620863990781631750	39.4784176043574344753379639995	49.2184563216946036702670828470	59.6795159441094188805359142004
74.8870067906951834448890413101	88.8264396098042275695104189989	103.499453895136580332223632536	118.899869163626464070691007678
139.040284426459849001591423516	157.913670417429737901351855998	177.520766813804649861717128073	197.857811193377198148830507786
222.932303617634156954562090522	246.740110027233965470862274997	271.281654272873333251355384624	296.554412135731364798058623390
326.563352932328456182245201466	355.305758439216910278041675996	384.781905102709358569870697386	414.989984259078215355284365727
449.933528518035526727300417270	483.610615653378572322890058994	518.021441011703061235909688476	553.164645838088553484610179799
593.042869655955288043670806203	631.654681669718951605407423992	671.000227622859697079667927572	711.078449733941523369057508986
755.891394783932962204831907870	799.437956488238048125593770990	843.718247936860052979737355857	888.731422469170525258762048428
938.479113475694210484385384309	986.960440108935861883449099988	1036.17549277098908978922348757	1086.12357854413106939063334625
k=4	k=5	k=6	k=7
26.3746164271633907701130803553	33.2174619142683688599231833505	40.7064658182003197420524327044	48.8311936436191988767343935418
70.8499989190958598621020031118	82.7192311014932799882174894334	95.2775725440371515166027398094	108.516358830155166631812575586
135.020708865970427320068775187	151.854874164068455262485930782	169.395449826099448170642951598	187.635838306952498541172857065
218.920189145663449665148278375	240.702906585416153631023868563	263.200854255008193287807812484	286.408957405342931321589916367
322.555116292544785687507490493	349.280079892073280672221377905	376.725399432167270819094589906	404.887083722287805440324251169
445.927564537333311368966180770	477.591818542018369994103529761	509.979675875275117798696474176	543.087927721502806525395832607
589.038351713452739372241445690	625.640324036469492050877317266	662.968088374682823239150562553	701.019014147859442122727160074
751.887853810112708794199708299	793.426630458528162048503333094	835.692744872819607407213491378	878.684003933551424395807159563
934.476263488699898250389903261	980.951276574572013504857538909	1028.15475988895977394826599236	1076.08485839956393318416725561
1136.80368781579493360782977767	1188.21456579725456251439220188	1240.35476859463231910539679471	1293.22270745096053825830444177

Contour plots of the eigenfunctions corresponding to the smallest eight eigenvalues (colored in red above). The second and eighth have the strongest singularity at the origin, \(r^{1/2}\).

The Problem

\[\int_\Omega a\nabla u\cdot\nabla v\,dx=\lambda\int_\Omega a uv\,dx\]

Let \(\Omega\subset \mathbb{R}^2\) be the unit disk, divided into four quadrants; and \(a=\kappa^2\;,\;\kappa>1\) in the first and third quadrant, while \(a=1\) in the second and fourth quadrants.

Eigenpairs of Type 1: \( (u_{km},\lambda_{km}) \) and \( (v_{km},\lambda_{km}) \)

\(\sigma_k=2k\;,\;\lambda_{km}=[j_m(\sigma_k)]^2\) \begin{align*} u_{km}&=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\cos(\sigma_k\,\theta)\quad,\quad k\geq 0\;,\; m\geq 1 \\ v_{km}&=a^{-1/2}J_{\sigma_k}(j_m(\sigma_k)\,r)\,\sin(\sigma_k\,\theta)\quad,\quad k\geq 1\;,\; m\geq 1 \end{align*}

Eigenpairs of Type 2: \( (w_{km},\mu_{km}) \)

\( \tau_k=2k+\frac{4}{\pi}\mbox{arccot}(\kappa) \;,\; \mu_{km}=[j_m(\tau_k)]^2\;,\; k\geq 0\;,\; m\geq 1\) \begin{align*} w_{km}&=a^{-1/2}J_{\tau_k}(j_m(\tau_k) r)\,g_k(\theta)\\ g_k(\theta)&=\begin{cases}- \cos(\tau_k(\pi/4-\theta))&,\,\theta\in[0,\pi/2)\\ - \sin(\tau_k(3\pi/4-\theta))&,\,\theta\in[\pi/2,\pi)\\ \cos(\tau_k(5\pi/4-\theta))&,\,\theta\in[\pi,3\pi/2)\\ \sin(\tau_k(7\pi/4-\theta))&,\,\theta\in[3\pi/2,2\pi)\\ \end{cases}\quad\mbox{ when }k\mbox{ is even}~,\\ g_k(\theta)&=\begin{cases}- \sin(\tau_k(\pi/4-\theta))&,\,\theta\in[0,\pi/2)\\ - \cos(\tau_k(3\pi/4-\theta))&,\,\theta\in[\pi/2,\pi)\\ \sin(\tau_k(5\pi/4-\theta))&,\,\theta\in[\pi,3\pi/2)\\ \cos(\tau_k(7\pi/4-\theta))&,\,\theta\in[3\pi/2,2\pi)\\ \end{cases}\quad\mbox{ when }k\mbox{ is odd} \end{align*}

Eigenpairs of Type 3: \( (z_{km},\nu_{km}) \)

\( \rho_k=2k-\frac{4}{\pi}\mbox{arccot}(\kappa) \;,\; \nu_{km}=[j_m(\rho_k)]^2\;,\; k\geq 1\;,\; m\geq 1\) \begin{align*} z_{km}&=a^{-1/2}J_{\rho_k}(j_m(\rho_k) r)\,h_k(\theta)\\ h_k(\theta)&=\begin{cases} \cos(\rho_k(\pi/4-\theta))&,\,\theta\in[0,\pi/2)\\ - \sin(\rho_k(3\pi/4-\theta))&,\,\theta\in[\pi/2,\pi)\\ - \cos(\rho_k(5\pi/4-\theta))&,\,\theta\in[\pi,3\pi/2)\\ \sin(\rho_k(7\pi/4-\theta))&,\,\theta\in[3\pi/2,2\pi)\\ \end{cases}\quad\mbox{ when }k\mbox{ is even}~,\\ h_k(\theta)&=\begin{cases} \sin(\rho_k(\pi/4-\theta))&,\,\theta\in[0,\pi/2)\\ - \cos(\rho_k(3\pi/4-\theta))&,\,\theta\in[\pi/2,\pi)\\ - \sin(\rho_k(5\pi/4-\theta))&,\,\theta\in[\pi,3\pi/2)\\ \cos(\rho_k(7\pi/4-\theta))&,\,\theta\in[3\pi/2,2\pi)\\ \end{cases}\quad\mbox{ when }k\mbox{ is odd} \end{align*}