Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

Transcription

1 Advaces i Pure Mathematics Published Olie Jauary 23 ( Poicaré Problem for Noliear Elliptic Equatios of Secod Order i Ubouded Domais Guochu We LMAM School of Mathematical Scieces Pekig Uiversity Beiig Chia wegc@mathpkueduc Received September 27 22; revised November 2 22; accepted November 22 ABSTRACT I [] I N Vekua propose the Poicaré problem for some secod order elliptic equatios but it ca ot be solved I [2] the authors discussed the boudary value problem for oliear elliptic equatios of secod order i some bouded domais I this article the Poicaré boudary value problem for geeral oliear elliptic equatios of secod order i ubouded multiply coected domais have bee completely ivestigated We first provide the formulatio of the above boudary value problem ad correspodig modified well posed-ess Next we obtai the represetatio theorem ad a priori estimates of solutios for the modified problem Fially by the above estimates of solutios ad the Schauder fixed-poit theorem the solvability results of the above Poicaré problem for the oliear elliptic equatios of secod order ca be obtaied The above problem possesses may applicatios i mechaics ad physics ad so o Keywords: Poicaré Boudary Value Problem; Noliear Elliptic Equatios; Ubouded Domais Formulatio of the Poicaré Boudary Value Problem Let D be a N -coected domai icludig the N ifiite poit with the boudary i C 2 Without loss of geerality we assume that D is a circular domai i the boudary cosists of circles N N r N ad D I this article the otatios are as the same i Refereces [-8] We cosider the secod order equatio i the complex form u F u u u G u u F Re Qu Au A2u A3 G G u u Q Q u u u A A u u 23 satisfyig the followig coditios Coditio C ) Q u w U A u w 23 are cotiuous i u w for almost every poit D U ad Q A 23 for D 2) The above fuctios are measurable i D for all cotiuous fuctios u w i D ad satisfy () Lp2 A u w D k 2 (2) Lp2 A 3 u w D k i which p p 2< p p k k are o-egative costats 3) The Equatio () satisfies the uiform ellipticity coditio amely for ay umber u ad w U U 2 the iequality F uwu F uwu 2 q UU2 for almost every poit D holds q is a o-egative costat 4) For ay fuctio u CD w W p2 D G u w satisfies the coditio G u u B u B2 u i which B B u u 2 satisfy the coditio L p2 B D k 2 (3) with a o-egative costat k Now we formulate the Poicaré boudary value problem as follows Problem P I the domai D fid a solutio u of Equatio () which is cotiuously differetiable i D ad satisfies the boudary coditio u cu c2 2 (4) ie Re u cu c2 i which is ay uit vector at every poit o D cos x icos y c are c ad 2 Copyright 23 SciRes

2 G C WEN 73 kow fuctios satisfyig the coditios C k C c k C c2 k2 (5) 2 k k2 are o-egative costats If cos ad c o is the outward ormal vector o the Problem P is the Dirichlet boudary value problem (Problem D) If cos ad a o the Problem P is the Neuma boudary value problem (Problem N) ad if cos ad c o the Problem P is the regular oblique derivative problem ie the third boudary value problem (Problem III or O) Now the directioal derivative may be arbitrary hece the boudary coditio is very geeral The iteger K arg 2π is called the idex of Problem P Whe the idex K Problem P may ot be solvable ad whe K the solutio of Problem P is ot ecessarily uique Hece we cosider the well-posedess of Problem P with modified boudary coditios Problem Q Fid a cotiuous solutio w u of the complex equatio w F u w w G u w F Re Qw Aw A2u A3 G B w B2 u (6) satisfyig the boudary coditio Re w c u c (7) 2 h ad the relatio u 2Re w N d b 2 (8) d N are appropriate real costats such that the fuctio determied by the itegral i (8) is sigle-valued i D ad the udetermied fuctio h is as stated i K N h N K h K N N K N h N K K m h Re hm ihm m id i which h N hm m K K are ukow real costats to be determied appropriately I additio for K the solutio w is assumed to satisfy the poit coditios Im a w a b 2K N K N J N K N K N 2 b J a N a N K N K N (9) are distict poits ad are all real costats satisfyig the coditios b 3 k J () for a o-egative costat 2 Estimates of Solutios for the Poicaré Boudary Value Problem First of all we give a prior estimate of solutios of Problem Q for (6) Theorem 2 Suppose that Coditio C holds ad ε = i (6) ad (7) The ay solutio w u of Problem Q for (6) satisfies the estimates i which k 3 C w D C u D M k (2) Lp 2 w w D M k (22) 2 mi 2 p M M q p k K D 2 k kk2 k3 k C w D C u D Proof Notig that the solutio w u of Problem Q satisfies the equatio ad boudary coditios w Re Qw Aw A G u w i D 3 Re w c2 h o Im a w a b J u b (23) (24) (25) accordig to the method i the proof of Theorem 43 Chapter II [2] or Theorem 22 [5] we ca derive that the solutio w satisfies the estimates 3 C w D M k (26) Lp 2 w w D M k 4 (27) Copyright 23 SciRes

3 74 G C WEN ad M M q p k K D 34 From (8) it follows that k k k 2 k3 Lp G D C 5 w D u D M C k3 (28) L p 2 u u DM C 5 w Dk3 (29) i which M 5 M5 p D is a o-egative costat Moreover it is easy to see that L p2 G D Lp2 B D C k u D Cu D L B D C p2 2 w D C w D (2) Combiig (26)-(2) the estimates (2) ad (22) are obtaied Theorem 22 Let the Equatio (6) satisfy Coditio C ad i (6)-(7) be small eough The ay solu- tio w u estimates of Problem Q for (6) satisfies the 6 C w D C u D M k (2) Lp 2 w w DLp 2 u DM7 k here p k are as stated i Theorem 2 M M q p k K D 67 Proof It is easy to see that w u the equatio ad boudary coditios 2 3 w Re Qw Aw Au A G D Re w cu c2 h Im a w a b J u b Moreover from (26) ad (27) we have C w DM3 k kc u D Lp 2 w w DM4 k kc u D ad from (28)-(2) it follows that C w DM3 k k M5C w Dk 3 Lp 2 w w DM4 k k M5C w Dk 3 (22) satisfies (23) (24) (25) (26) (27) If the positive costat is small eough such that kmm the the first iequality i (27) implies that D 3 C w k km3m5 k M (28) 2 k M k M k 3 8 Combiig (28) ad (28) we obtai C w D C u D M 5M8k M6k (29) which is the estimate (2) As for (22) it is easily derived from (29) ad the secod iequality i (27) ie Lp 2 w w DLp 2 u D M k k M C w Dk M C w Dk M4 k M5M8 km4 k M7k (22) 3 Solvability Results of the Poicaré Boudary Value Problem We first prove a lemma Lemma 3 If G u w satisfies the coditio stat- ed i Coditio C the the oliear mappig G: p2 C D C D L D defied by G G u w bouded is cotiuous ad p p B D Cu D L p2 G u w D L 2 B D C w D L 2 2 (3) Copyright 23 SciRes

4 G C WEN 75 p p 2 Proof I order to pr ove that the mappig G : C D C D L D p2 is cotiuous we Defied by G G u w choose ay sequece of fuctios w u w ucd 2 such that Cw w DCu u D as Similarly to Lemma 22 [5] we ca prove that C Gu wgu w possesses the property Lp2 C D as (32) Ad the iequality (3) is obviously true Theorem 32 Let the complex Equatio ( ) satisfy Coditio C ad the positive costat i (6) ad (7) is small eough ) Whe Problem Q for (6) has a solutio w u w u W p 2 D p 2 p p is a costat as state d before 2) Whe mi Problem Q for (6) has a solutio w u w W p 2 D provided that M 9 Lp2 A3 D C c2 b (33) J is sufficietly small 3) If F uww G uw satisfy the coditios ie Coditio C ad for ay fuctios w u CD 2 V L D there are p ad 2 F u w V F u 2 w2 V Re A w w 2 A2 u u2 G u wg u2 w2re B ww2 B 2uu2 (34) L 2 A D L p2 B p D k 2 is a sufficietly small positive costat th e the above solutio of Problem Q is uique Proof ) I this case the algebraic equatio for t is as follows M 6M7 Lp2 A3 DLp2 B Dt Lp2 B2 Dt L a2 b t (35) J{} M 6 M 7 are costats as stated i (2) ad (22) Because the Equatio (35) has a uique solutio t M Now we itroduce a bouded closed ad covex subset B * of the Baach space CD CD whose elemets are of the form wu satisfyig the coditio w ucd C w DCu D M (36) We choose a pair of fuctios w u B ad substitute it ito the appropriate positios of F uww G u w i (6) ad the boudary coditio (7) ad obtai F u w u w w G u w (37) w Re w c uc2 h (38) F uwuww Re Q uww wa uw w A u w ua u w 2 3 I accordace with the method i the proof of Theorem 25 [5] we ca prove that the boudary value problem (37) (38) ad (6) has a uique solutio w u Deote by wu T w u the map pig from w u to w u N otig that Lp2 Au 2 D Mk C cu Mk provided that the positive umber is sufficietly small ad otig that the coefficiets of complex Equatio (37) satisfy the same coditios as i Coditio C from Theorem 22 we ca obtai Cw DL w w DCu DL u D M6 M7M9 Lp2 B DCw D Lp2 B2 DCu D p2 p2 M6M7 L p2 A3 D C c2 b L p2 G D J M M M L B DM L B DM p2 p2 2 M (39) Copyright 23 SciRes

5 76 G C WEN This shows that T maps B * oto a compact subset i B * Next we verify that T i B * is a cotiuous operator I fact we arbitrarily select a sequece w u i B * such that C w w D C u u D as (3) By Lemma 3 we ca see that Lp2 A u w A u w D 23 as Moreover from w u Tw u w u Tw u (3) it is clear that w w u u is a solutio of Problem Q for the followig equatio w w F u w u w w F u w u w w (32) G u w G u w i D Re w w c u u h o awa Im w a J u u (33) ( 34) I accordace with the method i proof of Theorem 22 we ca obtai the estimate Cw w DL w w w w D p 2 Cu u DL u u D p 2 M L A u w u A u w p2 2 2 w Lp2 A3 u w A3 u w D Lp2 G u w G u D D C c u u (35) i which M M q p k K D From (3) (3) ad the above estimate we obtai Cw w DCu u D as O the basis of the Schauder fixed-poit theorem there exists a fuctio w u w u CD such that w u T w u ad from Th eorem sy to see t u W D ad 22 it is ea hat w p 2 w u is a solutio of Problem Q for the Equatio (6) ad the relatio (8) with the coditio u I additio if G u wrebw B2 u i D L p2 B D k 2 the the above solvability result still hold by usig the above similar method 2) Secodly we discuss the case: mi I this case (35) has the solutio t M provided that M 9 i (33) is small eough Now we cosider a closed ad covex subset B i the Baach space C D C D ie B w u C D Cw DCu DM (36) Applyig a method similar as before we ca verify that there exists a solutio w u Wp 2 D Wp 2 D of Problem Q for (6) with the coditio More over if 2 mi G u w ReBw B u i D Lp2 B D k = 2 Uder the same coditio we ca derive the above solvability result by the similar method 3) Whe G u w satisfies the coditio (34) we ca verify the uiqueess of solutios i this theorem I fact if w u w2 u2 are two solutios of Problem Q for the Equatio (6) the w u w w2 u u2 satisfies the equatio ad boudary coditios w Re Qw A B w A 2 B 2 u D (37) i which Re w cu h (38) Im a w a J (39) Q q Similarly to Theorem 22 we ca derive the followig estimates of the solutio w u for complex Equatio (37): 2 C w D C u D M k (32) Lp 2 w w D M3k (32) mi 2 p 23 k 2 kcu M M q p k K D are two o-egative costats over the estimate 2 3 D More- C w D k M C u D (322) ca be derived Provided that the positive costat is Copyright 23 SciRes

6 G C WEN 77 small eou gh such that 2 km 3 follows uu u ie u u from (322) it 2 2 i D This completes the proof of the theorem From the above theorem the ext result ca be derived Theorem 33 Uder the same coditios as i Theorem 32 the followig statemets hold ) Whe the idex K > N Problem P for () has N solvability coditios ad the solutio of Problem P depeds o 2K N 2 arbitrary real costats 2) Whe K N Problem P for () is solvable if 2N K solvability coditios are satisfied ad the solutio of Problem P depeds o K 2 arbitrary real costats 3) Whe K < Problem P for () is solvable uder 2N 2K coditios ad the solutio of Problem P depeds o arbitrary real costat Moreover we ca write dow the solvability coditios of Problem P for all other cases Proof Let the solutio w u of Problem Q for (6) be substituted ito the boudary coditio (7) ad the relatio (8) If the fuctio h ie h N K if K N h N hm m K if K ad d N the we have w u i D ad the fuctio w is ust a solutio of Problem P for () Hece the total umber of above equalities is ust the umber of solvability coditios as stated i this theorem Also ote that the real costats b i (8) ad b J i (9) are arbitrarily chose This shows that the geeral solutio of Problem P for () icludes the umber of arbitrary real costats as stated i the theorem REFERENCES [] I N Vekua Geeralied Aalytic Fuctios Pergamo Oxford 962 [2] G C We ad H Begehr Boudary Value Problems for Elliptic Equatios ad Systems Logma Scietific ad Techical Compay Harlow 99 [3] A V Bitsade Some Classes of Partial Differetial Equatios Gordo ad Breach New York 988 [4] G C We Coformal Mappigs ad Boudary Value Problems Traslatios of Mathematics Moographs 6 America Mathematical Society Providece 992 [5] G C We D C Che ad Z L Xu Noliear Complex Aalysis ad Its Applicatios Mathematics Moograph Series 2 Sciece Press Beiig 28 [6] G C We Approximate Methods ad Numerical Aalysis for Elliptic Complex Equatios Gordo ad Breach Amsterdam 999 [7] G C We Liear ad Quasiliear Complex Equatios of Hyperbolic ad Mixed Type Taylor & Fracis Lodo 22 doi:4324/ [8] G C We Recet Progress i Theory ad Applicatios of Moder Complex Aalysis Sciece Press Beiig 2 Copyright 23 SciRes