Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

Antunes, Pedro R. S.; Freitas, Pedro

doi:10.1007/s10957-011-9983-3

Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

Published: 10 January 2012

Volume 154, pages 235–257, (2012)
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Pedro R. S. Antunes^1,2 &
Pedro Freitas^2,3

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Abstract

We perform a numerical optimization of the first ten nontrivial eigenvalues of the Neumann Laplacian for planar Euclidean domains. The optimization procedure is done via a gradient method, while the computation of the eigenvalues themselves is done by means of an efficient meshless numerical method which allows for the computation of the eigenvalues for large numbers of domains within a reasonable time frame. The Dirichlet problem, previously studied by Oudet using a different numerical method, is also studied and we obtain similar (but improved) results for a larger number of eigenvalues. These results reveal an underlying structure to the optimizers regarding symmetry and connectedness, for instance, but also show that there are exceptions to these preventing general results from holding.

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Acknowledgements

P.R.S.A. was partially supported by FCT, Portugal, through the scholarship SFRH/BPD/47595/2008 and the project PTDC/MAT/105475/2008 and by Fundação Calouste Gulbenkian through program Estímulo à Investigação 2009. Both authors were partially supported by FCT’s project PTDC/MAT/101007/2008.

Author information

Authors and Affiliations

Department of Mathematics, Universidade Lusófona de Humanidades e Tecnologias, Av. do Campo Grande, 376, 1749-024, Lisbon, Portugal
Pedro R. S. Antunes
Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, 1649-003, Lisbon, Portugal
Pedro R. S. Antunes & Pedro Freitas
Department of Mathematics, Faculdade de Motricidade Humana, TU Lisbon, Lisbon, Portugal
Pedro Freitas

Authors

Pedro R. S. Antunes
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Pedro Freitas
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Corresponding author

Correspondence to Pedro Freitas.

Additional information

Communicated by Enrique Zuazua.

Appendix

i	λ ₅		λ ₆		λ ₇		λ ₈
i	a _i	b _i	a _i	b _i	a _i	b _i	a _i	b _i
0	0.55438079		0.540969934		0.555917408		0.52402981
1	0.013238729	0.0057015487	−0.0254618108	0.20040578	−0.0396571523	0.0142910591	−0.133073983	−0.237327033
2	−0.077358408	0.092876235	−0.0441431399	−0.0456353393	−0.0378443687	−0.069786321	0.0524340446	0.0850167772
3	−0.0051545105	−0.0024247559	−0.0573759273	0.023490155	−0.0159573574	0.059561064	−0.0123987019	−0.0376173185
4	0.014513074	0.080309088	−0.0106613347	−0.0482010972	−0.0563539206	0.0490047495	−0.0260590608	0.00874912734
5	−0.0060402322	0.0077267846	0.0138861761	0.000120165574	−0.00279806649	−0.00185896562	−0.00308877567	−0.00973987438
6	0.0080050385	0.0023040320	0.000823401662	−0.00116176266	0.028061714	0.00907415168	−0.00634415879	0.014659245
7	−0.00041462756	0.0055027319	0.000493117601	0.000994308524	−0.00621703077	0.00398846543	−0.00742943607	0.00144421808
8	0.011735634	−0.0056190380	0.00160548125	−0.00108261442	0.00163688539	0.00365127087	0.0144983946	0.00753703381
9	0.0039973580	0.0034615840	−0.0000526885146	0.000351429203	0.00321047409	−0.00372583508	−0.00642345426	−0.013566282
10	−0.00022151852	−0.0026638205	0.000112772803	−0.0000539188865	0.0066201419	−0.00279887602	−0.00391602052	0.0101569665
11	0.0027374112	0.00075580032	−0.000110774901	0.000119156464	−0.00362005727	0.000298194234	0.00186757844	−0.00127916948
12	−0.0017708460	−0.0030278106	−0.000109752183	−0.000168572897	−0.00258236878	−0.00312578819	0.0024586174	0.00171021319
13	0.00096320890	−0.0020044397	7.12030548×10⁻⁶	−0.0000117806279	0.00193116396	0.0000971958322	−0.00172422653	−0.00153156433
14	−0.000035727810	0.00044155670	7.24155168×10⁻⁶	−3.84022077×10⁻⁶	−0.00131702223	−0.000360018056	0.00181204829	0.00094827332
15	−0.00053585540	−0.0017723963	−4.96715037×10⁻⁶	−3.61610015×10⁻⁶	−0.00172076406	0.000688255022	−0.00166054256	−0.00127832356
16	−0.00012345948	0.00096474668	3.08139935×10⁻⁷	−2.03467268×10⁻⁷	−0.00115709151	0.000109810932	0.000431561493	0.00130351922
17	−0.0013188383	−0.00070898727	1.33654316×10⁻⁷	1.43346737×10⁻⁷	0.00149459761	0.000610453469	0.0000346589834	0.000803396577
18	0.00022774519	0.000042899804	−1.13982504×10⁻⁷	5.93369227×10⁻⁸	−0.000305459945	0.00101611823	0.00188952975	−0.00135359719
19	−0.00050075176	0.000026338167			−0.000427708258	−0.000303869144	−0.00221559806	−0.000232152954
20	−0.00016570749	0.00018779596			0.000678645943	0.000276590019	0.00103793066	0.000830350291
21					0.000569909436	−0.000046937353	−0.000313475612	−0.000209535319
22					0.0000654203457	0.0000144614045	0.000450916946	−0.000139709455
23					−0.000351309674	−0.000371289997	−0.000211391115	0.0001928562
24					0.000351733859	−0.000136480239	0.000217311751	−0.000318264501

i	λ ₉		λ ₁₀		λ ₁₁		λ ₁₂
i	a _i	b _i	a _i	b _i	a _i	b _i	a _i	b _i
0	0.550655939		0.5595999481423		0.556134818		0.560041023
1	−0.0998751448	0.0773421852	0.0138865410279	−0.008909759517606	−0.0368731391	0.0239315354	−0.0379604707	0.0121487257
2	0.0550157453	−0.0430957094	0.002833319704712	0.002227574982416	−0.0182053322	0.0378034906	0.0387729478	0.0507913179
3	0.0399594874	0.0593875966	−0.008349328761967	0.0957550910192	−0.00715126258	−0.101591534	−0.0015923326	0.00266165893
4	0.00142349384	0.0425900486	0.005723472594822	−0.002967862015974	−0.0297483526	−0.0418656294	−0.0146793258	0.0490107094
5	−0.0139749255	−0.0408617143	0.002652563387632	0.002447234406475	0.0241573412	0.011609453	0.00402780473	−0.0090959291
6	−0.00367223638	0.0134695479	−0.004599950115557	0.02627159459515	0.0164630521	−0.00243660354	0.0172020883	−0.00377988974
7	0.0082532258	0.00455014924	−0.002226983087317	0.000837603292585	−0.00634085209	0.00454033198	0.000684280074	0.00988820653
8	−0.00895830418	0.00176185763	0.00158099064617	0.00129569319721	−0.00412733089	0.0136032862	0.0176838961	0.0101877652
9	0.00783739536	0.00272856592	0.000470007408465	−0.00146767096302	−0.00119221343	−0.00583869131	−0.00650954246	−0.00231789514
10	0.00224548719	−0.00470960161	−0.00032089848269	−0.0000433152786976	−0.000348634021	0.0023618601	0.00215815089	0.000591548356
11	−0.00333457231	0.00120461234	−0.000417096748991	−0.000549411311543	−0.00458553797	−0.000155987857	0.00268858702	−0.000746677809
12	0.00200923924	0.00134340334	0.00118106640714	−0.00326484110276	0.00156859838	−0.00100931829	0.00277871992	−0.00316449624
13	−0.00128480877	−0.00273791879	0.0000474565496931	0.0000106513746558	0.00246569204	−0.000345244088	−0.00204072395	0.00161161335
14	0.000462592585	0.00386907619	−0.000325293746937	−0.000447970381423	−0.000674880003	−0.00290011162	−0.0000851755712	−0.00180079941
15	0.00220397703	−0.00328595287	−0.0000233520418177	−2.21572997176×10⁻⁶	0.000577306217	0.0013066555	0.000479952574	−0.00218761547
16	−0.00159326685	0.000960521718	0.000145484964248	0.0000270002808451	−0.000130890779	−0.000367319041	−0.00119840099	−0.00172733708
17	0.000330803919	0.000303830978	0.0000346822452809	0.0000929323504196	−0.00153045766	−0.0000950422371	0.00130704997	0.00124434852
18	−0.000225563872	−0.00129473302	−0.000323698743562	0.000580384511426	0.000111369245	0.0000164522889	−0.00102042771	−0.00100033718
19	−0.000189994632	0.000379332968	5.83949533850×10⁻⁶	−3.49905606658×10⁻⁶	0.000181076275	−0.000228614786	−0.000755371705	0.0000336785202
20	0.00114449161	0.000603512001	0.0000728114477851	0.000158698077368	0.0000332405057	−0.00075412169	−0.000423177546	0.000174586669
21	−0.00107566805	−0.000685698552	−6.75766392730×10⁻⁹	8.82905652608×10⁻⁹	0.0000453530838	0.000211630126	0.000529771728	−0.000195846058
22			−4.71286357457×10⁻⁹	−1.34781200234×10⁻⁹	0.000790026567	0.00042327352	−0.000711361356	0.000555750456
23			4.91209220912×10⁻⁹	−1.56660058965×10⁻⁹	−0.000395412526	−0.0000237025238	−0.000200059087	0.000361447102
24			1.34638456165×10⁻⁸	−1.42615694109×10⁻⁸	−0.000221380759	0.000184318601	6.73027374×10⁻⁶	0.000121028574
25			−2.01122360346×10⁻⁹	−3.63972837821×10⁻¹⁰
26			−5.44330996748×10⁻⁹	−6.68514460573×10⁻⁹
27			1.73465493720×10⁻⁹	−1.98827703541×10⁻⁹
28			1.26990184897×10⁻⁹	6.02497710742×10⁻¹⁰
29			−6.50409422882×10⁻¹⁰	5.20844861007×10⁻¹⁰
30			−4.33794173095×10⁻⁹	2.75350252810×10⁻⁹

i	λ ₁₃		λ ₁₄		λ ₁₅
i	a _i	b _i	a _i	b _i	a _i	b _i
0	0.551503971		0.557612527		0.559420408
1	−0.0338396841	−0.114953381	0.0207678515	−0.0631536572	6.04658369×10⁻⁶	0.0000124733917
2	0.0303279427	−0.0574495473	0.058792891	−0.00649772139	−0.0000161762498	5.76792466×10⁻⁶
3	−0.0130257251	−0.0745641579	0.00531111557	0.0725253769	−0.00258516561	0.098348305
4	0.00554729211	0.0470925562	−0.0238412727	0.00370821936	0.0000123512599	−0.000190815291
5	−0.000623410989	0.0156226091	−0.00423481014	0.0177123045	−0.000180152753	−0.000020711265
6	0.017655584	0.0236085722	0.0103355869	0.00370104625	−0.0272911266	−0.00107299012
7	0.0157809243	0.0041362414	0.00304837882	−0.014888849	0.0000811807213	0.000119491243
8	−0.0102727122	0.00626879621	−0.00901708416	−0.00469316105	0.0000417425306	−0.000256107706
9	0.00478087159	0.0101645762	−0.00164226506	0.0112119165	0.00153912059	−0.0158989451
10	−0.00238858367	−0.00266695979	0.00355556561	0.00270396021	−0.0000614558942	−0.000447492308
11	−0.00105832356	−0.00408004353	0.000321931966	−0.00139053897	0.000866001007	0.00027348823
12	−0.000322435582	−0.00132277381	0.000904139462	−0.000968101304	−0.00316306275	−0.000317748954
13	−0.000213314175	−0.00418096453	−0.000540135261	0.00109132915	−0.000144984378	−0.0000655946379
14	0.00272342258	0.000515668982	0.00225122256	0.00138225898	0.0000138295682	0.0000408821376
15	0.00227560059	−0.00387180384	0.00168277384	−0.00245914572	0.000674764382	−0.00440846194
16	−0.00134117822	0.000421056294	−0.0020040944	−0.00149449634	0.000187045993	−0.0000389099159
17	0.00206300976	0.00168123748	−0.00133504032	0.00121246955	0.000295365946	0.000120717313
18	−0.000134937278	−0.000594329311	0.000932926429	0.000698600595	0.00201216141	0.000363000545
19	−0.000198444012	0.000832646684	0.000619420867	−0.000348513148	0.0000688156095	−0.0000196772322
20	−0.00102360784	0.0000520444273	−0.000149699453	−0.000278429144	−0.0000745226863	0.00023283672
21	−0.001672364	0.000149383067	−0.0000699271742	−5.4335581×10⁻⁶	0.000139260778	−0.000810604043
22	−0.0000358586874	0.000530041321	0.0000435046984	0.0000790199395	0.0000853310383	−0.0000589423176
23	−0.000396920563	−0.00174332362	0.000122963842	−0.0000894017094	−9.68881282×10⁻⁶	−0.0000137876809
24	−0.000877487074	0.000231158959	−0.000096357168	−0.000130817975	0.000859421574	0.000254606817
25	0.00081526229	−0.0000801225237			0.000192873826	0.00041027021
26	−0.0000126186682	−0.000184605499			−0.0000242683214	0.000159546939
27	0.000719525244	0.00045645044			−0.0000380641833	0.0000291315401
28	0.0000402334404	0.0000752523567
29	−0.000231434094	0.000612933714
30	0.000248027728	0.000375568407

i	μ ₃		μ ₄		μ ₅
i	a _i	b _i	a _i	b _i	a _i	b _i
0	0.549954737		0.554098864		0.560519732
1	−0.000578925467	0.122335927	0.00784197933	0.00418017833	2.70058702×10⁻⁶	−7.43462854×10⁻⁷
2	0.0268566607	0.000218577103	−0.0567066976	−0.085617596	−2.82208992×10⁻⁶	0.0891304954
3	0.000496273706	−0.119430145	0.0000940623335	0.00223713046	−2.99864606×10⁻⁶	3.78746736×10⁻⁶
4	0.0393289299	0.000347273779	−0.0418915905	0.1008615	0.0176267143	−1.07708404×10⁻⁶
5	0.000137109558	0.0106100598	−0.00119189439	0.000890469706	2.41171388×10⁻⁶	2.88093629×10⁻⁶
6	0.00283111302	−0.000494884753	−0.000628688812	−0.000171550392	−3.02951992×10⁻⁷	0.000287510816
7	0.000178562726	0.00858917695	0.000993259124	0.000551021581	−1.89820251×10⁻⁶	1.35592844×10⁻⁶
8	−0.0044258337	−0.000117746227	0.00172088603	0.00189395041	−0.0000537048883	−3.03014709×10⁻⁷
9	−0.000153351823	−0.00245969402	0.0000211410532	0.000121738707	1.11865906×10⁻⁸	1.5845649×10⁻⁸
10	0.000262693154	0.00011309051	0.0000861800292	−0.000464999058	−7.95639895×10⁻¹⁰	1.16121376×10⁻⁸
11	0.0000273742248	−0.000637837391	−2.92218066×10⁻⁶	3.16893374×10⁻⁶	−9.58692935×10⁻⁹	7.27334368×10⁻⁹
12	0.000277665815	0.000118370769	0.000038215104	−1.87411565×10⁻⁶	−1.01220124×10⁻⁸	−2.3451735×10⁻¹⁰
13	4.89182689×10⁻⁷	−1.22498345×10⁻⁶	5.19443562×10⁻⁷	−6.26013908×10⁻⁶	−8.39746687×10⁻¹⁰	−1.23147374×10⁻⁹
14	−7.17958187×10⁻⁷	−5.83425372×10⁻⁷	−2.72365156×10⁻⁸	4.12250759×10⁻⁸
15	9.84379966×10⁻⁷	−1.28371466×10⁻⁶	−3.8438191×10⁻⁷	1.2958845×10⁻⁷
16	1.0499127×10⁻⁶	3.93269146×10⁻⁷	−1.20147834×10⁻⁷	−8.02744085×10⁻⁸
17	−5.21573808×10⁻⁸	1.3948447×10⁻⁷
18	4.29340708×10⁻⁷	−7.32876443×10⁻⁸

i	μ ₆		μ ₇		μ ₈
i	a _i	b _i	a _i	b _i	a _i	b _i
0	0.544423006		0.561303447		0.563472327
1	0.154549722	0.074592044	−0.0296105015	0.00929288081	−0.0135281387	−0.0125695822
2	−0.0238297433	−0.0134221544	0.00292713876	0.000461527094	−0.00100221576	0.000360922605
3	0.0465889872	0.0735117289	−0.00581168985	0.00629595371	−0.000758596993	−0.00183183995
4	0.00798942924	0.0640407746	0.000521812939	0.000923350669	0.0108989574	−0.00829182165
5	−0.00631739792	−0.018003381	−0.00154983934	0.0094039286	−0.000820543007	−0.000308128932
6	−0.0137357316	0.0204107763	−0.00501494717	0.0700259021	0.000874631017	−0.00117284521
7	−0.0216354835	0.0151204116	−0.00243458512	−0.012367072	−0.00888157387	−0.00150292924
8	−0.00906075347	0.00115927695	0.00113180953	0.000671399157	0.000674798064	0.000729850385
9	−0.008733436	0.000345375567	0.00256031464	0.00298770473	−0.0000147809707	0.000267662456
10	0.00229280469	−0.00114925445	0.00279992286	−0.000175531225	−0.00054844287	−0.000335737527
11	0.00315240014	0.00284257911	0.00354013286	0.00287589877	0.00119052948	−0.000881687946
12	−0.00163249012	0.00100770067	0.0137409909	0.000577609805	0.000195303616	0.0000981460855
13	−0.0000192004444	0.00315093287	−0.0052461102	0.00145050804	0.000174689747	0.000173203814
14	−0.00187726514	0.0026655375	0.00146193752	−0.0000610893421	−0.00126202335	−0.000458787859
15	−0.00216069733	0.00147185804	0.00022791128	−0.0021485341	0.0000822630097	0.000478005166
16	−0.00226807794	0.00129214283	−0.0000970608746	0.000143359763	1.09927346×10⁻⁶	0.0000300340072
17	−0.00096819417	−0.000488799805	0.000144426536	−0.00032671668	−9.92661542×10⁻⁶	−2.66527939×10⁻⁶
18	−0.000118656413	−0.00018293087	0.0000407362083	−0.000176673501	0.0000515695341	−9.26540504×10⁻⁶
19	−0.000192492127	0.0000493303754	−0.0000273822824	0.000171666015	−7.89252523×10⁻⁶	−6.26345568×10⁻⁶
20	0.000263760246	0.000350170164	0.0000432423872	−0.000127265244	2.41000428×10⁻⁶	−2.3749484×10⁻⁶
21	1.16558991×10⁻⁶	4.96712883×10⁻⁶			−5.85227252×10⁻⁶	−6.79087762×10⁻⁶
22	−1.63244689×10⁻⁶	6.47331955×10⁻⁶			−3.33045158×10⁻⁶	7.49271783×10⁻⁶
23	−3.4729084×10⁻⁶	5.95322586×10⁻⁶			1.88193258×10⁻⁶	−3.98305519×10⁻⁷
24	−4.96630548×10⁻⁶	1.72968518×10⁻⁶			−2.36175431×10⁻¹⁸	2.51061885×10⁻⁷
25	5.12755036×10⁻⁸	2.14945715×10⁻⁷
26	−3.07397112×10⁻⁸	2.19743806×10⁻⁷
27	−6.99669598×10⁻⁸	1.77349366×10⁻⁷

i	μ ₉		μ ₁₀
i	a _i	b _i	a _i	b _i
0	0.546107642		0.557982648
1	0.132184562	0.0832860664	−0.0107226947	−0.00560550488
2	−0.0422564242	−0.081406976	−0.00243459937	0.00272185071
3	−0.00288494405	0.0494657212	−0.106247514	−0.0216645017
4	−0.00990364257	0.0141660798	0.00371146609	−0.00230659129
5	0.044032033	−0.02072678	0.0000381990189	0.00245899729
6	0.0162253823	0.00284538146	0.0299771487	0.017040206
7	−0.00684008357	−0.00453911102	−0.0033437228	0.00348204544
8	0.0113448501	0.0332865286	0.000299992113	−0.00319195161
9	−0.00329408749	0.0181081652	−0.0217413819	−0.0152507074
10	−0.001807493	0.000642929037	0.000428329649	0.00273289835
11	−0.00649512127	0.00171952483	0.000587136389	−0.0000355914864
12	0.00379245547	0.00021410323	0.0000998793688	−0.000710946153
13	−0.000881363531	−0.00148150927	0.00168461816	−0.00141128391
14	−0.00288415573	−0.0106440172	−0.000916519918	0.000996942438
15	0.000669513161	−0.0000464470469	0.00151666691	0.00459881141
16	0.00163822411	−0.000542216743	−0.00150433091	0.000444760945
17	0.0087891035	−0.00109922112	0.00109118785	−0.000873399337
18	0.000888529246	0.000976724264	−0.00103745245	−0.00391413716
19	0.000561401536	0.00191176013	0.000136745573	0.000451314701
20	0.000172718181	0.00178646218	−0.000328686271	0.0000923832995
21	−0.0000607718052	−0.00115913985	0.0000607676685	0.000247990695
22	−0.00224226772	0.000812563001	0.000109497006	−0.0000372289029
23	−0.0026768554	0.0000114857432	−0.0000756102401	0.0000118837984
24	0.00107005965	−0.000253133326	−0.000067632927	0.000143418041

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Antunes, P.R.S., Freitas, P. Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians. J Optim Theory Appl 154, 235–257 (2012). https://doi.org/10.1007/s10957-011-9983-3

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Received: 21 September 2011
Accepted: 20 December 2011
Published: 10 January 2012
Issue Date: July 2012
DOI: https://doi.org/10.1007/s10957-011-9983-3

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Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

Abstract

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Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction

The eigenvalues of the Laplacian with Dirichlet boundary condition in $$\mathbb {R}^2$$ are almost never minimized by disks

Optimization of the First Dirichlet Laplacian Eigenvalue with Respect to a Union of Balls

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Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

Abstract

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Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction

The eigenvalues of the Laplacian with Dirichlet boundary condition in $$\mathbb {R}^2$$ are almost never minimized by disks

Optimization of the First Dirichlet Laplacian Eigenvalue with Respect to a Union of Balls

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Acknowledgements

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