UNIVERSAL BOUNDS FOR EIGENVALUES OF FOURTH-ORDER WEIGHTED POLYNOMIAL OPERATOR ON DOMAINS IN COMPLEX PROJECTIVE SPACES

Transcription

1 wwwrresscom/volmes/vol7isse/ijrras_7 df UNIVERSAL BOUNDS FOR EIGENVALUES OF FOURTH-ORDER WEIGHTED POLYNOIAL OPERATOR ON DOAINS IN COPLEX PROJECTIVE SPACES D Feng & L Ynl * Scool of emcs nd Pyscs Scence Jngc Unversy of Tecnology Hbe Jngmen PRCn Scool of Elecronc nd Informon Scence Jngc Unversy of Tecnology Hbe Jngmen PRCn *E-ml: @qqcomcom ABSTRACT In s er we sdy e egenvles roblem of for-order weged olynoml oeror nd ge generl neqly on comc Remnnn mnfolds By sng s generl neqly we obn nversl bonds for e - egenvle n erms of e lower egenvles ndeendenly of e domns Keywords: For-order weged olynoml oeror egenvles nversl bonds comlex roecve sce 000 R Sbec Clssfcon: 35P5 53C40 INTRODUCTION Le be bonded domn n n n -dmensonl comlee Remnnn mnfold Le be e Llcn oeror cng on fncons on nd consder e followng egenvle roblem for e brmonc oeror n 0 on were denoes e owrd n norml vecor feld of dscree secrm 0 < < I s nown s egenvle roblem s n n were ec egenvle s reeed w s mllcy Wen R Pyne-Póly-Wenberger x [8] n 956 roved 8n n In 984 Hle nd Ye [6] srengened nd roved In 006 Ceng-Yng [] gve e followng mc sronger neqly 3 8 n n 4 Tese neqles re clled nversl neqles becse ey do no nvolve domn deendence In s er we consder e followng egenvle roblem of for-order weged olynoml oeror 5

2 Feng & Ynl Unversl Bonds for Egenvles of For-Order 5 were s osve nd connos fncon on nd e consns b 0 In 00 Sn-Cen [9] obn nversl neqles for roblem 5 n n -dmensonl Eclden sce ey roved were n 4 n n 4 B 6 In s er we wll consder e roblem 5 on bonded domns n e n-dmensonl comlex roecve sce CP n 4 Ten we obn Teorem Assme Ω be bonded domn n n-dmensonl comlex roecve sce CP n 4 le λ be e egenvle of e egenvle roblem 5 nd 7 From e eorem we cn ge e followng weer b more exlc neqly Corollry Under e ssmon of e eorem we ve 8 A generl neqly In s secon we wll nrodce generl neqly wc s ly ey role n e roof of e mn resls Lemm Le be n n-dmensonl comc Remnnn mnfold w bondry emy ossbly 6

3 Feng & Ynl Unversl Bonds for Egenvles of For-Order Le λ be e egenvle of e egenvle roblem of for-order weged olynoml oeror w weg sc nd be e oronorml egenfncon corresondng o s Ten for ny C 4 we ve were roof Le for ny neger were en we ve 0 nd 0 from e Ryleg-Rz neqly we ge b By drecly comon we ve 3 nd 4 By 3 nd 4 we ve b 5 were + Becse of 0 we cn ge b - - b 7

4 Feng & Ynl Unversl Bonds for Egenvles of For-Order 8 b 6 were b By nd 6 we ve b 7 Usng negron by rs we ve dv dv dv dv + 8 wc mles 9 We lso ve 0 nd Combnng 9- we ge

5 Feng & Ynl Unversl Bonds for Egenvles of For-Order 3 I follows from 7 nd 3 Seng 4 en - - nd 5 By 4 5 nd Scwrz neqly we ge were s ny osve consn Smmng over from o n 6 we ve 6 9

6 Feng & Ynl Unversl Bonds for Egenvles of For-Order 30 7 Le s come 8 nd 9 Snce we ve 0 Inrodcng 8-0 no 7 we ge Hence s re s comlees e roof of Lemm 3 PROOF OF THE AIN RESULTS In s secon we wll gve e roof of Teorem nd Corollry Frsly we nrodce lemm Lemm 3[4] Le be bonded domn n n -dmensonl comlex roecve sce 4 n CP P ere re some fncons } { n g ssfy

7 Feng & Ynl Unversl Bonds for Egenvles of For-Order n g n g n n g g 6n n g 4n 0 3 roof of Teorem Tng g n nd smmng over from o n g n n Becse of x nd Tng 33 no 3 nd nocng g g we ve g n we cn ge 33 4n n g n n g Snce b en we ve nd by Scwrz neqly we ve 4n g we ve g b b 35 From 35 nd 36 we cn ge s s qdrc neqly of solvng we obn 3

8 Feng & Ynl Unversl Bonds for Egenvles of For-Order Seng wc mly Snce A A we ve From 3 36 nd 37 we cn ge n g g n g g g g 4 4n n nd by e defnon of we cn ge 3

9 Feng & Ynl Unversl Bonds for Egenvles of For-Order 30 Inrodcng 39 nd 30 no 34 we obn 4n 8 n A 6n n 3 In 30 ng we cn ge 3 33 wc mles 33

10 Feng & Ynl Unversl Bonds for Egenvles of For-Order Ts comlees e roof of Teorem 34 roof of Corollry By 3 we obn 4n 8 n A Tng n n 35 we ve Ts comlees e roof of Corollry ACKNOWLEDGENT: Te reserc wor s sored by Key Lborory of Aled emcs of Hbe Provnce nd Te reserc roec of Jngc Unversy of Tecnology REFERENCES [] S Asbg Isoermerc nd nversl neqles for egenvles In: Dves EB Sflov Yeds Secrl eory nd geomery Ednbrg 998 London Soc Lecre Noes Cmbrdge Unversy Press Cmbrdge [] D Cen Q Ceng Exrnsc esmes for egenvles of e Llce oeror J Soc Jn [3] Q Ceng HC Yng Ineqles for egenvles of clmed le roblem Trns Amer Soc [4] Q Ceng HC Yng Ineqles for egenvles of Llcn on domns nd comc comlex yersrfces n comlex roecve sces J Soc Jn [5] GN Hle RZ Ye Ineqles for egenvles of e brmonc oeror Pcfc J [6] GN Hle H Proer Ineqles for egenvles of e Llcn Indn Unv J [7] E Hrrel J Sbbe On rce neqles nd e nversl egenvle esmes for some rl dffernl oerors Trns Am Soc [8] LE Pyne G P o ly HF Wenberger On e ro of consecve egenvles J Pys [9] H Sn D Cen Esmes for egenvles of for-order weged olynoml oeror Ac Sc 3B [0] Q Wng C X Ineqles for egenvles of clmed le roblem Clc Vr PDE [] HC Yng An esme of e dfference beween consecve egenvles rern IC/9/60 of ICTP Trese 99 34