Về một số phương trình elliptic và hyperbolic phi tuyến suy biến

B GIO DC V O TO TRNG I HC S PHM H NI * DNG TRNG LUYN V MT S PHNG TRèNH ELLIPTIC V HYPERBOLIC PHI TUYN SUY BIN LUN N TIN S TON HC H NI - 2017 B GIO DC V O TO TRNG I HC S PHM H NI * DNG TRNG LUYN V MT S PHNG TRèNH ELLIPTIC V HYPERBOLIC PHI TUYN SUY BIN Chuyờn ngnh: Phng trỡnh vi phõn v tớch phõn Mó s: 62.46.01.03 LUN N TIN S TON HC Ngi hng dn khoa hc: GS.TSKH Nguyn Minh Trớ H NI - 2017 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi Cỏc kt qu ny c lm di s hng dn ca GS TSKH Nguyn Minh Trớ Cỏc kt qu lun ỏn vit chung vi thy hng dn u ó c s nht trớ ca thy hng dn a vo lun ỏn Cỏc kt qu lun ỏn l trung thc v cha tng c cụng b cỏc cụng trỡnh ca cỏc tỏc gi khỏc Nghiờn cu sinh: Dng Trng Luyn LI CM N Lun ỏn c thc hin v hon thnh ti B mụn Gii tớch, Khoa Toỏn - Tin, Trng i hc S phm H Ni, di s hng dn ca GS TSKH Nguyn Minh Trớ Thy ó dn dt tỏc gi lm quen vi nghiờn cu khoa hc tỏc gi cũn l hc viờn cao hc Ngoi nhng ch dn v mt khoa hc s ng viờn v lũng tin tng ca thy dnh cho tỏc gi luụn l ng lc giỳp tỏc gi tin tng v say mờ nghiờn cu khoa hc Vi tm lũng tri õn sõu sc, tỏc gi xin by t lũng bit n chõn thnh v sõu sc nht i vi thy Tỏc gi xin trõn trng gi li cm n n Ban Giỏm hiu, Phũng sau i hc, Ban Ch nhim Khoa Toỏn - Tin, Trng i hc S phm H Ni, c bit l cỏc thy giỏo, cụ giỏo B mụn Gii tớch, Khoa Toỏn - Tin, Trng i hc S phm H Ni, cỏc thy giỏo, cụ giỏo Phũng Phng trỡnh vi phõn, Vin Toỏn hc, ó luụn giỳp , ng vin, to mụi trng hc nghiờn cu thun li cho tỏc gi Tỏc gi xin cm n Ban Giỏm hiu, cỏc anh ch em Khoa T nhiờn, Trng i hc Hoa L ó to mi iu kin thun li giỳp tỏc gi quỏ trỡnh hc nghiờn cu v hon thnh lun ỏn Tỏc gi xin trõn trng cm n qu NAFOSTED ó ti tr cho tỏc gi sut quỏ trỡnh hc nghiờn cu sinh Li cm n sau cựng, xin dnh cho gia ỡnh ca tỏc gi, nhng ngi ó dnh cho tỏc gi tỡnh yờu thng trn vn, tng ngy chia s, ng viờn tỏc gi vt qua mi khú khn hon thnh lun ỏn Mc lc Trang Li cam oan Li cm n Mc lc Mt s quy c v kớ hiu M u Tng quan 10 Chng1 MT S KIN THC CHUN B 17 1.1 1.2 Toỏn t v mt s khụng gian hm 17 1.1.1 Toỏn t 17 1.1.2 Mt s khụng gian hm 19 1.1.3 Mt s tớnh cht 20 Tp hỳt ton cc v tớnh cht 23 1.2.1 Mt s nh ngha 23 1.2.2 Mt s tớnh cht 26 Chng2 S TN TI NGHIM V TNH CHNH QUY CA NGHIM CA BI TON BIấN I VI PHNG TRèNH ELLIPTIC SUY BIN 2.1 28 Mt s nh lớ v s tn ti nghim yu 28 2.1.1 nh lớ v s tn ti nghim yu 29 2.1.2 nh lớ v s tn ti nghim yu khụng õm 41 2.2 Tớnh chớnh quy ca nghim ca bi toỏn biờn elliptic suy bin 44 Chng3 TP HT TON CC I VI PHNG TRèNH HYPERBOLIC TT DN CHA TON T ELLIPTIC SUY BIN MNH TRONG MIN B CHN 52 3.1 S tn ti v nht ca nghim tớch phõn 53 3.1.1 t bi toỏn 53 3.1.2 S tn ti v nht ca nghim tớch phõn 54 3.2 () ì L2 () S tn ti hỳt ton cc S(k ,k2 ),0 3.3 ỏnh giỏ s chiu fractal ca hỳt ton cc 61 69 Chng4 TP HT TON CC I VI PHNG TRèNH HYPERBOLIC TT DN CHA TON T GRUSHIN TRấN TON KHễNG GIAN 4.1 4.2 82 S tn ti nht ca nghim tớch phõn 83 4.1.1 t bi toỏn 83 4.1.2 S tn ti v nht ca nghim tớch phõn 84 S tn ti hỳt ton cc Sk2 (RN ) ì L2 (RN ) 86 Kt lun v kin ngh 111 Cỏc kt qu t c 111 Kin ngh mt s nghiờn cu tip theo 111 Danh mc cụng trỡnh khoa hc ca tỏc gi liờn quan n lun ỏn 113 Ti liu tham kho 113 MT S QUY C V K HIU Trong ton b lun ỏn, ta thng nht mt s kớ hiu nh sau: RN khụng gian vect thc N chiu R+ cỏc s thc khụng õm R+ cỏc s thc dng |x| chun Euclid ca phn t x khụng gian RN C k () khụng gian cỏc hm kh vi liờn tc n cp k C0 () khụng gian cỏc hm kh vi vụ hn cú giỏ compact Lp () khụng gian cỏc hm ly tha bc p kh tớch Lebesgue H ã, ã khụng gian i ngu ca khụng gian Banach H i ngu gia H v H (ã, ã)H tớch vụ hng khụng gian H Id ỏnh x ng nht hi t yu phộp nhỳng liờn tc phộp nhỳng compact Vol() o Lebesgue ca khụng gian RN x y z Toỏn t Laplace theo bin x RN1 : x = Toỏn t Laplace theo bin y RN2 : y = Toỏn t Laplace theo bin z RN3 : z = N1 i=1 N2 j=1 N3 l=1 x2i yj2 zl2 M U Lớ chn ti Lớ thuyt phng trỡnh vi phõn o hm riờng c nghiờn cu u tiờn cỏc cụng trỡnh ca J DAlembert (1717-1783), L Euler (17071783), D Bernoulli (1700-1782), J Lagrange (1736-1813), P Laplace (1749-1827), S Poisson (1781-1840) v J Fourier (1768-1830), nh l mt cụng c chớnh mụ t c hc cng nh mụ hỡnh gii tớch ca Vt lớ Vo gia th k XIX vi s xut hin cỏc cụng trỡnh ca Riemann, lớ thuyt phng trỡnh vi phõn o hm riờng ó chng t l mt cụng c thit yu ca nhiu ngnh toỏn hc Cui th k XIX, H Poincarộ ó ch mi quan h bin chng gia lớ thuyt phng trỡnh vi phõn o hm riờng v cỏc ngnh toỏn hc khỏc Sang th k XX, lớ thuyt phng trỡnh vi phõn o hm riờng phỏt trin vụ cựng mnh m nh cú cụng c gii tớch hm, c bit l t xut hin lớ thuyt hm suy rng S L Sobolev v L Schwartz xõy dng Nghiờn cu cỏc phng trỡnh, h phng trỡnh elliptic tng quỏt v phng trỡnh hyperbolic ó úng vai trũ rt quan trng lớ thuyt phng trỡnh vi phõn Hin cỏc kt qu theo hng ny ó tng i hon chnh Cựng vi s phỏt trin khụng ngng ca toỏn hc cng nh khoa hc cụng ngh nhiu bi toỏn liờn quan ti trn ca nghim ca cỏc phng trỡnh, h phng trỡnh khụng elliptic v phng trỡnh hyperbolic tt dn suy bin ó xut hin Cú mt s lp phng trỡnh, ú cú lp phng trỡnh elliptic suy bin v phng trỡnh hyperbolic tt dn suy bin, mt khớa cnh no ú cng cú mt s tớnh cht ging vi phng trỡnh elliptic v phng trỡnh hyperbolic tt dn cha toỏn t Tuy nhiờn cỏc kt qu t c cho cỏc phng trỡnh elliptic v hyperbolic tt dn suy bin cũn ớt, cha y Vi cỏc lớ nờu trờn chỳng tụi ó chn ti nghiờn cu cho lun ỏn ca mỡnh l V mt s phng trỡnh elliptic v hyperbolic phi tuyn suy bin Mc ớch nghiờn cu Ni dung : Nghiờn cu bi toỏn biờn elliptic suy bin cha toỏn t vi cỏc ni dung sau: - Nghiờn cu s tn ti nghim yu ca bi toỏn; - Tớnh chớnh quy ca nghim yu Ni dung : Nghiờn cu phng trỡnh hyperbolic tt dn cha toỏn t elliptic suy bin mnh b chn vi cỏc ni dung sau: - Nghiờn cu s tn ti v nht nghim tớch phõn; - Nghiờn cu s tn ti hỳt ton cc; - ỏnh giỏ s chiu fractal ca hỳt ton cc Ni dung : Nghiờn cu phng trỡnh hyperbolic tt dn cha toỏn t Grushin ton khụng gian vi cỏc ni dung sau: - Nghiờn cu s tn ti v nht nghim tớch phõn; - Nghiờn cu s tn ti hỳt ton cc i tng v phm vi nghiờn cu i tng nghiờn cu ca lun ỏn l xột bi toỏn biờn v bi toỏn biờn giỏ tr ban u cú cha toỏn t elliptic suy bin N := j=1 xj j2 xj Phng phỏp nghiờn cu nghiờn cu s tn ti nghim yu ca bi toỏn chỳng tụi s dng phng phỏp bin phõn nghiờn cu tớnh chớnh quy ca nghim chỳng tụi s dng nh lớ nhỳng kiu Sobolev v mt s bt ng thc nghiờn cu s tn ti nht ca nghim tớch phõn chỳng tụi s dng phng phỏp na nhúm (xem [53, 59]) nghiờn cu dỏng iu tim cn ca nghim, chỳng tụi s dng cỏc cụng c v phng phỏp ca lớ thuyt h ng lc vụ hn chiu (xem [13,14,21,22,52,58,62,74]), núi riờng l phng phỏp phng trỡnh nng lng v phng phỏp ỏnh giỏ phn uụi ca nghim chng minh s chiu fractal ca hỳt ton cc l b chn chỳng tụi s dng phng phỏp qu o (xem [51, 55]) Cỏc kt qu t c v ý ngha ca ti Lun ỏn ó t c nhng kt qu chớnh sau õy: i vi bi toỏn biờn elliptic suy bin a v chng minh c s tn ti nghim yu ca bi toỏn vi mt s iu kin ca s hng phi tuyn.V cng chng minh c tớnh chớnh quy ca nghim õy l ni dung ca Chng i vi phng trỡnh hyperbolic tt dn cha toỏn t elliptic suy bin mnh b chn: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc v ỏnh giỏ c s chiu fractal ca hỳt õy l ni dung ca Chng i vi phng trỡnh hyperbolic tt dn cha toỏn t Grushin ton khụng gian: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc õy l ni dung ca Chng Cỏc kt qu ca lun ỏn l mi, cú ý ngha khoa hc, v gúp phn Vy hm f (X, u) tha iu kin (4.2) Ta cú f (X, 0) = L2 (R2 ), nờn f (X, u) tha iu kin (4.3) Vi iu kin (4.4), ta chn C1 = 14 , g1 (X) = 4(|X|14 +1) L1 (R2 ) C2 |X|4 +1 Tht vy, ta tỡm C1 v g2 (X) = C2 F (X, u) C1 f (X, u)u + |X|4 + u(1u2 ) u2 u4 |X|4 +1 C1 |X|4 +1 + C2 |X|4 +1 |u| 1, u2 u4 14 + C2 |u| 1, |X| +1 C1 u u |u| 1, u2 u4 uC1 u2 + C2 u2 u4 C1 u u2 + C2 C1 u4 + C1 u2 C2 tha |u| 1, C1 ta chn C1 = |u| 1, 4 u4 + C1 u2 C2 + |u| 1, ú C2 tha |u| 1, 41 u2 C2 u2 C2 + |u| 1, nờn ta chn C2 = 41 Khi ú ta cú 1 F (X, u) f (X, u)u , X R2 , u R 4 4(|X| + 1) Ta cú |X|4 + 1 |X|4 + u2 u4 0, X R2 , |u| 1, u2 u4 0, X R2 , |u| 109 Nờn F (X, u(X)) 0, vi mi u S1/2 (R2 ) Vy hm f (X, u) tha iu kin (4.5) Do ú f (X, u) tha cỏc iu kin (4.2)(4.5) Nờn ỏp dng nh lớ 4.2.7, ta cú Vớ d 4.2.8 tn ti mt hỳt ton cc 2 2 (R2 )ìL2 (R2 ) S AS1/2 1/2 (R ) ì L (R ) i vi na nhúm S(t) sinh bi Vớ d 4.2.8 KT LUN CHNG Trong chng ny, chỳng tụi nghiờn cu bi toỏn hyperbolic tt dn cha toỏn t Grushin ton khụng gian, vi s hng phi tuyn tha mt s tớnh cht Cỏc kt qu t c bao gm: 1) Chng minh c s tn ti nht ca nghim tớch phõn ca bi toỏn (nh lớ 4.1.5 ) 2) Chng minh c s tn ti hỳt ton cc ca bi toỏn (nh lớ 4.2.7) Cỏc kt qu ca Chng l s m rng cỏc kt qu tng ng trc ú i vi phng trỡnh hyperbolic tt dn cha toỏn t elliptic ton khụng gian [27, 33] cho toỏn t Grushin v hm phi tuyn Phng phỏp c s dng õy l phng phỏp c lng uụi nghim cho hỡnh cu suy bin tng ng vi toỏn t Grushin 110 KT LUN V KIN NGH Cỏc kt qu t c Trong lun ỏn ny, chỳng tụi nghiờn cu s tn ti nghim, tớnh chớnh quy ca nghim ca bi toỏn biờn cú cha phng trỡnh elliptic suy bin b chn cú biờn trn v s tn ti nghim ton cc, hỳt ton cc ca bi toỏn biờn giỏ tr ban u i vi phng trỡnh hyperbolic tt dn cú cha toỏn t elliptic suy bin Cỏc kt qu chớnh t c lun ỏn bao gm: i vi bi toỏn biờn elliptic suy bin a v chng minh c s tn ti nghim yu ca bi toỏn vi mt s iu kin ca s hng phi tuyn, v tớnh chớnh quy ca nghim i vi phng trỡnh hyperbolic tt dn cha toỏn t elliptic suy bin mnh b chn: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc liờn thụng compact khụng gian S(k () ì L2 () ,k2 ),0 v chng minh c s chiu fractal ca hỳt l hu hn i vi phng trỡnh hyperbolic tt dn cha toỏn t Grushin ton khụng gian: Chng minh c s tn ti v nht ca nghim tớch phõn Chng minh c s tn ti ca hỳt ton cc khụng gian Sk2 (RN ) ì L2 (RN ) Kin ngh mt s nghiờn cu tip theo Bờn cnh cỏc kt qu t c lun ỏn, mt s m liờn quan cn c tip tc nghiờn cu nh: 111 Nghiờn cu iu kin tn ti nghim ca cỏc bi toỏn biờn núi trờn khụng b chn, b chn vi iu kin biờn Dirichlet khụng thun nht Nghiờn cu s tn ti hỳt hm phi tuyn ph thuc thi gian nh hỳt lựi, hỳt u v cỏc tớnh cht ca hỳt Nghiờn cu s tn ti nghim v dỏng iu tim cn nghim 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