Bậ GIO DệC V TRìNG O TO I HC Sì PHM H NậI BềI KIM MY P DệNG PHìèNG PH•P GIƒI T•CH NGHI–N CÙU MËT SÈ B€I TO•N ELLIPTIC SUY BI˜N LUŠN •N TI˜N Sž TO•N HÅC H Nëi, 2019 Bậ GIO DệC V TRìNG O TO I HC Sì PHM H NậI BềI KIM MY P DệNG PHìèNG PHP GII TCH NGHIN CU MậT Sẩ BI TON ELLIPTIC SUY BIN LUN N TIN S TON HC Chuyản ng nh: ToĂn giÊi tẵch M số: 46 01 02 Ngữới hữợng dăn khoa hồc PGS TS Cung Thá Anh H€ NËI, 2019 LÍI CAM OAN Tỉi xin cam oan Ơy l cổng trẳnh nghiản cựu cừa tổi CĂc kát quÊ viát chung vợi cĂc tĂc giÊ khĂc ãu  ữủc sỹ nhĐt trẵ cừa cĂc ỗng tĂc giÊ ữa v o luên Ăn CĂc kát quÊ trẳnh b y luên Ăn l mợi v chữa tứng ữủc cổng bố bĐt bĐt kẳ cổng trẳnh cõa kh¡c H Nëi, th¡ng 02 n«m 2019 NCS Bũi Kim My i LI CM èN Luên Ăn ữủc ho n th nh tÔi Bở mổn GiÊi tẵch, Khoa ToĂn, trữớng Ôi hồc Sữ phÔm H Nởi 2, dữợi sỹ hữợng dăn tên tẳnh chu Ăo cừa PGS.TS Cung Thá Anh TĂc giÊ xin b y tọ lỏng kẵnh trồng v vổ biát ỡn tợi ThƯy, ngữới  truyãn Ôt kián thực, kinh nghiằm hồc têp v nghiản cựu khoa hồc, nh hữợng tĂc giÊ tiáp cên hữợng nghiản cựu thới sỹ, thú v v cõ ỵ nghắa TĂc giÊ xin chƠn th nh cÊm ỡn PGS.TS KhuĐt Vôn Ninh, PGS.TS Nguyạn Nông TƠm, TS TrƯn Vôn Bơng (tr÷íng HSP H Nëi 2), TS Phan Qc H÷ng (tr÷íng H Duy TƠn, Nđng)  ởng viản v cho tĂc giÊ nhỳng gõp ỵ, kinh nghiằm nghiản cựu khoa håc gióp t¡c gi£ ho n th nh luªn ¡n n y T¡c gi£ cơng xin c£m ìn c¡c Th¦y, Cỉ v c¡c Anh, Chà nghi¶n cùu sinh ð X¶mina GiÊi tẵch, Khoa ToĂn, trữớng HSP H Nởi v Xảmina cừa Bở mổn GiÊi tẵch, Khoa ToĂn-Tin trữớng Ôi hồc Sữ phÔm H Nởi  tÔo mởt mổi trữớng hồc têp, nghiản cựu khoa hồc sổi nời v thƠn thi»n, gióp t¡c gi£ ho n th nh luªn ¡n n y T¡c gi£ xin b y tä láng c£m ỡn án Ban GiĂm hiằu trữớng Ôi hồc Duy TƠn  hộ trủ mởt phƯn kinh phẵ tĂc giÊ ho n th nh luªn ¡n n y Líi c£m ìn sau còng, t¡c gi£ xin b y tä láng biát ỡn tợi gia ẳnh, nhỳng ngữới thƠn  luổn bản, tin tững v cho tĂc giÊ ởng lỹc tinh thƯn tĂc giÊ ho n th nh luên ¡n n y T¡c gi£ cơng xin c£m ìn c¡c anh ch em, bÔn b  giúp ù, ởng viản º t¡c gi£ ho n th nh luªn ¡n n y ii Mưc lưc LÍI CAM i OAN LÍI CƒM ÌN ii MÖC LÖC MËT SÈ K HIU THìNG DềNG TRONG LUN N MÐ †U Lch sỷ vĐn ã v lẵ chồn · t i Mửc ẵch nghiản cựu 16 ối tữủng v phÔm vi nghiản cựu 16 Ph÷ìng ph¡p nghi¶n cùu 17 Kát quÊ cừa luên Ăn 17 CĐu trúc cừa luên Ăn 18 Ch÷ìng KI˜N THÙC CHU‰N BÀ 20 1.1 To¡n tû -Laplace 20 1.2 C¡c khỉng gian h m v ph²p nhóng 23 1.3 Mët v i k¸t quÊ cừa lẵ thuyát im tợi hÔn 29 1.4 Mët sè i·u kiằn tiảu chuân trản số hÔng phi tuyán 34 Ch÷ìng Sĩ TầN TI NGHIM CếA PHìèNG TRNH ELLIPTIC SUY BIN NÛA TUY˜N T•NH 37 2.1 °t b i to¡n 37 2.2 Sỹ tỗn tÔi nghiằm yáu khổng tƯm th÷íng 40 2.3 Tẵnh a nghiằm cừa nghiằm yáu 48 Ch÷ìng Sĩ TầN TI V KHặNG TầN TI NGHIM CếA H HAMILTON SUY BI˜N 53 3.1 °t b i to¡n 53 3.2 Sỹ khổng tỗn tÔi nghiằm cờin dữỡng 57 3.3 Sỹ tỗn tÔi cừa mởt dÂy vổ hÔn nghi»m y¸u 64 Chữỡng NH L KIU LIOUVILLE CHO H BT THC ELLIPTIC SUY BI˜N •NG 75 4.1 °t b i to¡n 75 4.2 ành l½ kiºu Liouville cho tr÷íng hđp p; q > 76 4.3 ành l½ kiºu Liouville cho tr÷íng hđp p; q > 82 K˜T LUŠN V€ KI˜N NGHÀ 86 DANH MệC CặNG TRNH KHOA HC CếA TC GIƒ 88 T€I LI›U THAM KHƒO 89 MậT Sẩ K HIU THìNG DềNG TRONG LUŠN •N N khỉng gian vectì thüc N chi·u; ; chuân Euclide khổng gian R ; ối ngău giúa X v X ; R jj h i N (;) tẵch vổ hữợng khổng gian Hilbert X; Q = 2Q số chiãu thuƯn nhĐt cừa khổng gian RN ; số mụ tợi hÔn php nhúng kiu Sobolev; Q C0 ( ) khæng gian c¡c h m khÊ vi vổ hÔn cõ giĂ compact ; L ( ) khỉng gian c¡c h m lơy thøa bêc p khÊ tẵch Lebesgue ; ( ; ) Lp tẵch vổ hữợng khổng gian L ( ); k k Lp ! chu©n khỉng gian L ( ); hởi tử mÔnh; * hởi tử yáu; ,! php nhóng li¶n tưc; ,!,! ph²p nhóng compact; p p p N 1;p toĂn tỷ suy bián mÔnh W ( ) P i := @xi ( i @xi ); =1 khỉng gian h m dòng º nghi¶n cùu b i to¡n Ch÷ìng 2, 3; k k1;p W 2;p ( ) 1;p chu©n khỉng gian W ( ); khỉng gian h m dòng º nghi¶n cùu b i toĂn Chữỡng 3; 2;p k k2;p chuân khổng gian W giĂ tr riảng Ưu tiản cừa toĂn tỷvợi iãu kiằn biản A s ( ); Dirichlet thuƯn nhĐt; s lụy thứa bêc s cừa toĂn tỷ A vỵi mi·n x¡c ành D(A ): MÐ †U Lch sỷ vĐn ã v lẵ chồn ãti Nhiãu phữỡng trẳnh Ôo h m riảng loÔi elliptic g-n vợi viằc nghiản cựu trÔng thĂi dứng cừa cĂc quĂ trẳnh tián hõa vêt lẵ, hõa hồc, cỡ hồc v sinh hồc Mt khĂc, nhiãu lợp phữỡng trẳnh elliptic phi tuyán quan trồng cụng xuĐt phĂt tứ cĂc b i toĂn cừa hẳnh hồc vi phƠn (xin xem cĂc chuy¶n kh£o [5, tr.75-138], [25, tr.309-367], [33, tr.13-216], [58, tr.7-68, 251-266], [74, tr.1-68]) Vẳ vêy, viằc nghiản cựu nhỳng lợp phữỡng trẳnh n y cõ ỵ nghắa quan trồng khoa håc v cỉng ngh» Mët m°t vi»c nghi¶n cùu cĂc phữỡng trẳnh elliptic thúc ây v cung cĐp ỵ tững cho sỹ phĂt trin cĂc cổng cử v kát quÊ cừa nhiãu chuyản ng nh giÊi tẵch nhữ Lẵ thuyát cĂc khổng gian h m, GiÊi tẵch h m phi tuyán, Php tẵnh bián phƠn, M°t kh¡c, sü ph¡t triºn cõa c¡c chuy¶n ng nh n y dăn án nhỳng tián bở lợn lẵ thuyát phữỡng trẳnh elliptic Chẵnh vẳ vêy lẵ thuyát phữỡng trẳnh elliptic  v ang thu hút ữủc sỹ quan tƠm cừa nhiãu nh khoa hồc trản thá giợi Trong nhỳng nôm tr lÔi Ơy, sỹ tỗn tÔi nghiằm, sỹ khổng tỗn tÔi nghiằm, v cĂc tẵnh chĐt nh tẵnh cừa nghiằm  ữủc nghiản cựu cho nhiãu lợp phữỡng trẳnh v hằ phữỡng trẳnh elliptic nỷa tuyán tẵnh trữớng hủp khổng suy bián hoc suy bián yáu Tuy nhiản, cĂc kát quÊ tữỡng ựng ối vợi cĂc lợp phữỡng trẳnh v hằ phữỡng trẳnh elliptic trữớng hủp suy bián mÔnh văn cỏn ẵt Nguyản l tẵnh suy bián mÔnh cừa hằ gƠy nhỳng khõ khôn lỵn v· m°t to¡n håc, häi ph£i câ nhúng ỵ tững mợi tiáp cên Chng hÔn, nhỳng khõ khôn gƠy thiáu cĂc nh lẵ nhúng cƯn thiát, thiáu cĂc kát quÊ cƯn thiát vã tẵnh chẵnh quy nghiằm cừa b i toĂn tuyán tẵnh tữỡng ựng, thiáu cĂc kát quÊ vã nguyản lẵ cỹc tr, B¥y gií, ta chån > cho + v > thäa m¢n p+1 q+1 1 + = 1, thäa = p + 1: Tứ õ, ta lĐy l số liản hủp cừa , tùc l m¢n w = = q + 1: p dửng bĐt u = (uv) + ¯ng thùc Young, ta câ v = uq+1 + vp+1 (4.17) uq+1 + vp+1: Tø (4.16) v sû dưng b§t ¯ng thùc Cauchy, ta câ 2r ur v r ur v + jr ur vj r ur v + p v pu p r u p r v u v v u 2 r ur v + ujr uj + v jr vj = (4.18) 2w jr wj : Tø (4.15), (4.17), v (4.18), ta thu ÷đc w 1jr wj2: w+w (4.19) Náu ta t w = g thẳ ta câ N N Xi w = (g ) = X @x =1 N i X ( (x)@x g ) = i i @ x i ( (x)2g@x g) i i=1 i i 2 = 2i (x)[(@xi g) + g@xixi g] = 2g g + 2jr gj ; (4.20) =1 v 22 2 jr wj = jr g j = j2gr gj = 4g jr gj : Thay (4.20) v (4.21) v o (4.19), ta thu ÷đc 2g g + 2jr gj + g 4g2jr gj2; 2g 83 (4.21) iãu n y tữỡng ữỡng vợi g Tø gi£ thi¸t p; q > v g2 : thọa mÂn iãu kiằn pq > 1; nản ta câ = + = p+q+2 < 1; p + q + pq + p + q + hay > 1; i·u n y tữỡng ữỡng vợi > 1: Hỡn nỳa, tứ gi£ thi¸t = + Q p+1 q+1 Q ta thu ÷đc Do â, ta câ < Q Q Q : Q ; ¡p döng H» qu£ 4.1 ð trản ta thu ữủc g 0; v tứ õ u v 0: Chựng minh cừa nh lẵ 4.2ữủc ho n th nh Kát luên Chữỡng Trong chữỡng n y, chúng tổi nghiản cựu cĂc nh lẵ kiu Liouville cho N h» b§t ¯ng thùc elliptic chùa to¡n tû to n khỉng gian R ; N 2: C¡c k¸t quÊ Ôt ữủc bao gỗm: 1) Chựng minh ữủc nh lẵ kiu Liouville ( nh lẵ 4.1) cho hằ bĐt ¯ng thùc elliptic suy bi¸n sè mơ p; q > 1: 84 2) Chựng minh ữủc nh lẵ kiu Liouville ( nh lẵ 4.2) cho hằ bĐt ng thực elliptic suy bi¸n sè mơ p; q > (vợi miãn bián thiản cừa p; q khĂc vợi trữớng hđp p; q > 1) Nâi ri¶ng, u = v v p = q, ta thu ÷đc c¡c ành lẵ kiu Liouville cho bĐt ng thực elliptic chựa toĂn tû Khi p; q > 1, c¡c k¸t qu£ nhên ữủc trản l tối ữu (nhữ  biát trữớng hủp toĂn tỷ Laplace) CĂc kát quÊ chữỡng n y m rởng cĂc kát quÊ tữỡng ựng trữợc õ cho toĂn tỷ Laplace [30, 49, 66], v cho to¡n tû Grushin [23] 85 K˜T LUŠN V KIN NGH CĂc kát quÊ Ôt ữủc Luên Ăn nghiản cựu sỹ tỗn tÔi, khổng tỗn tÔi v tẵnh a nghiằm cừa mởt số lợp phữỡng trẳnh, hằ phữỡng trẳnh, v hằ bĐt ng thực elliptic suy bián chựa toĂn tỷ : CĂc kát quÊ Ôt ữủc cừa luên Ăn bao gỗm: 1) Chựng minh ữủc sỹ tỗn tÔi v tẵnh a nghiằm cừa nghiằm yáu cừa phữỡng trẳnh elliptic suy bián nỷa tuyán tẵnh miãn b chn, số hÔng phi tuyán cõ tông trững dữợi tợi hÔn v khổng thọa mÂn iãu kiằn (AR) 2) Chựng minh ữủc sỹ khổng tỗn tÔi nghiằm cờ in dữỡng miãn t-hẳnh v tẵnh a nghiằm cõa nghi»m y¸u cõa h» elliptic suy bi¸n nûa tuy¸n tẵnh dÔng Hamilton miãn b chn 3) Thiát lêp ữủc mởt số nh lẵ kiu Liouville cho bĐt ng thực v hằ N bĐt ng thực elliptic suy bián to n khỉng gian R : Ki¸n nghà mởt số vĐn ã nghiản cựu tiáp theo Bản cÔnh cĂc kát quÊ Â Ôt ữủc luên Ăn, mởt số vĐn ã m cƯn ữủc tiáp tửc nghiản cựu bao gỗm: 1) Nghiản cựu cĂc iãu kiằn tỗn tÔi nghiằm cừa phữỡng trẳnh, hằ phữỡng trẳnh elliptic suy bián nõi trản trữớng hủp số hÔng phi tuyán cõ tông trững tợi hÔn Nghiản cựu tẵnh chẵnh quy cừa nghiằm yáu 2) Nghiản cựu cĂc ựng dửng cừa cĂc nh lẵ kiu Liouville nhữ Ănh 86 giĂ hơng số phê qu¡t, ¡nh gi¡ k¼ dà cõa nghi»m, ¡nh gi¡ ë suy gi£m, sü bòng nê cõa nghi»m, 87 DANH MƯC CỈNG TRœNH KHOA HÅC CÕA T•C GIƒ [1] C.T Anh and B.K My, (2016), Existence of solutions to Laplace equations without the Ambrosetti-Rabinowitz condition, Com-plex Var Elliptic Equ 61 No.1, 137-150 (SCIE) [2] C.T Anh and B.K My, (2016), Liouville-type theorems for elliptic inequalities involving the -Laplace operator, Complex Var Elliptic Equ 61 No.7, 1002-1013 (SCIE) [3] C.T Anh and B.K My, 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