Bất đẳng thức tích chập suy rộng KontorovichLebedev – Fourier và ứng dụng

B GIO DC V O TO TRNG I HC BCH KHOA H NI - PHM VN HONG BT NG THC TCH CHP SUY RNG KONTOROVICH-LEBEDEV - FOURIER V NG DNG LUN N TIN S TON HC H Ni - 2017 B GIO DC V O TO TRNG I HC BCH KHOA H NI - PHM VN HONG BT NG THC TCH CHP SUY RNG KONTOROVICH-LEBEDEV - FOURIER V NG DNG LUN N TIN S TON HC Chuyờn ngnh: Toỏn gii tớch Mó ngnh: 62460102 TP TH HNG DN KHOA HC: PGS TS NGUYN XUN THO PGS TS TRNH TUN H Ni - 2017 MC LC MC LC Chng KIN THC C S 1.1 Khụng gian Lebesgue Lp () v Lp (; ) 1.2 Bin i tớch phõn Fourier 1.2.1 nh ngha v tớnh cht 1.2.2 Bt ng thc tớch chp Fourier 1.3 Bin i tớch phõn Kontorovich-Lebedev 1.3.1 Tớch chp Kontorovich-Lebedev 1.3.2 Tớch chp suy rng Kontorovich-Lebedev 1.4 Trng nhiu x súng õm, súng in t vi biờn hỡnh nún trũn 1.4.1 Biu din trng nhiu x súng õm 1.4.2 Biu din th Debye ca trng nhiu x súng in t 15 15 17 17 18 20 24 25 27 27 32 LI CAM OAN MT S K HIU DNG TRONG LUN N M U Chng BIN I TCH PHN KIU TCH CHP SUY RNG KONTOROVICH-LEBEDEV-FOURIER 2.1 2.2 2.3 Tớch chp suy rng Kontorovich-Lebedev - Fourier sine - Fourier cosine 2.1.1 nh ngha 2.1.2 Tớnh cht toỏn t 2.1.3 Tớnh khụng cú c ca khụng Bin i tớch phõn kiu tớch chp suy rng Kontorovich-Lebedev - Fourier Phng trỡnh vi-tớch phõn liờn quan tớch chp suy rng 34 34 34 36 42 44 49 Chng BT NG THC TCH CHP SUY RNG KONTOROVICHLEBEDEV 53 3.1 Bt ng thc i vi tớch chp suy rng Kontorovich-Lebedev - Fourier 53 53 57 62 62 66 68 74 Chng MT S NG DNG 4.1 Trng nhiu x súng õm vi tr khỏng dng nún 4.1.1 Biu din trng nhiu x súng õm theo tớch chp suy rng 4.1.2 Tớnh b chn ca trng nhiu x súng õm trờn cỏc khụng gian Lp (R+ ), p 4.1.3 c lng ti lõn cn nh nún 4.2 Th Debye ca trng nhiu x súng in t 4.2.1 Xỏc nh hm ph ca th Debye trng nhiu x 4.2.2 Biu din th Debye trng nhiu x theo tớch chp suy rng Kontorovich-Lebedev - Fourier 4.2.3 c lng a phng 4.3 Phng trỡnh dng parabolic 4.3.1 Phng trỡnh parabolic tuyn tớnh liờn quan tớch chp suy rng Kontorovich-Lebedev 4.3.2 Phng trỡnh parabolic phi tuyn liờn quan tớch chp Kontorovich-Lebedev KT LUN DANH MC CễNG TRèNH CễNG B CA LUN N TI LIU THAM KHO 82 82 3.2 3.3 3.1.1 Bt ng thc kiu Young 3.1.2 Bt ng thc kiu Saitoh Bt ng thc i vi tớch chp Kontorovich-Lebedev 3.2.1 Bt ng thc kiu Young 3.2.2 Bt ng thc kiu Saitoh 3.2.3 Bt ng thc kiu Saitoh ngc Phng trỡnh vi-tớch phõn liờn quan n toỏn t Bessel 83 84 87 88 91 91 92 93 94 102 106 107 108 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi, di s hng dn ca cỏc thy PGS TS Nguyn Xuõn Tho v PGS TS Trnh Tuõn Tt c cỏc kt qu c trỡnh by Lun ỏn l hon ton trung thc v cha tng c cỏc tỏc gi khỏc cụng b bt k cụng trỡnh no H Ni, Ngy 10 thỏng 10 nm 2017 Thay mt th hng dn PGS.TS Nguyn Xuõn Tho Tỏc gi Phm Vn Hong LI CM N Lun ỏn c thc hin v hon thnh di s hng dn tn tỡnh ca cỏc thy PGS TS Nguyn Xuõn Tho v PGS TS Trnh Tuõn Tỏc gi xin c by t s kớnh trng v lũng bit n sõu sc n cỏc thy, nhng ngi ó dn dt tỏc gi t nhng bc i u tiờn trờn ng nghiờn cu, ng viờn tỏc gi vt qua khú khn quỏ trỡnh lm NCS Tỏc gi xin c gi li cm n chõn thnh n cỏc thy cụ v cỏc thnh viờn Seminar Gii tớch-i s trng HKHTN-HQGHN, Seminar Gii tớch Trng HBK H Ni, nhng ngi luụn gn gi, giỳp v to iu kin thun li tỏc gi hc v trao i chuyờn mụn Tỏc gi xin c by t lũng bit n sõu sc n thy GS TSKH V Kim Tun (i hc West Georgia, M), ngi ó luụn ng viờn v cho tỏc gi nhiu ý kin quý bỏu quỏ trỡnh hc Trong thi gian lm NCS ti Trng i hc Bỏch khoa H Ni, tỏc gi ó nhn c nhiu tỡnh cm cng nh s giỳp t cỏc thy cụ B mụn Toỏn c bn, cỏc thy cụ Vin Toỏn ng dng v Tin hc Tỏc gi xin c chõn thnh by t lũng bit n sõu sc n cỏc thy cụ Tỏc gi xin c gi li cm n n TS Nguyn Thanh Hng (HSP H Ni), TS Nguyn Hong Thoan (Vin Vt lý k thut, HBK H Ni), TS Tng Duy Hi (Khoa Vt lý, HSP H Ni) v nhng giỳp quỏ trỡnh lm NCS Tỏc gi xin c by t lũng bit n n Lónh o S Giỏo dc v o to H Ni, Ban Giỏm hiu v cỏc ng nghip thuc T Toỏn-Tin, Trng THPT Kim Liờn ó to iu kin thun li quỏ trỡnh tỏc gi c hc tp, cụng tỏc v hon thnh Lun ỏn Cui cựng, tỏc gi xin c by t lũng bit n sõu nng n gia ỡnh, b m, v con, cỏc anh ch em Nim tin yờu v hi vng ca mi ngi l ngun ng viờn v l ng lc to ln tỏc gi vt qua mi khú khn sut quỏ trỡnh hc tp, nghiờn cu v hon thnh Lun ỏn Tỏc gi MT S K HIU DNG TRONG LUN N R l tt c cỏc s thc R+ = {x R, x > 0} R = {x R, < x < }, vi l s thc dng C l tt c cỏc s phc (z) l phn thc ca s phc z (z) l phn o ca s phc z C0 (R+ ) l khụng gian cỏc hm liờn tc trờn R+ v trit tiờu ti vụ cựng vi chun sup C 2,1 (R2+ ) l khụng gian cỏc hm hai bin u(x, t) kh vi liờn tc cp theo bin x trờn R+ v kh vi liờn tc theo bin t trờn R+ Ap,q (t) l biu thc cú dng (xem trang 16) Ap,q (t) = p t pq (1 t) p1 1q q 1 1 (1 t p ) p (1 t q ) q F l bin i tớch phõn Fourier (xem trang 17) Fc l bin i tớch phõn Fourier cosine (xem trang 17) Fs l bin i tớch phõn Fourier sine (xem trang 18) l toỏn t Laplace-Beltrami trờn mt cu S (xem trang 29) E l trng súng in (xem trang 32) H l trng súng t (xem trang 32) D1 l toỏn t vi phõn bc vụ hn c xỏc nh bi cụng thc (xem trang 44) d2 d x x x N dx dx D1 = lim + N (2k 1)2 k=1 D l toỏn t vi phõn bc vụ hn c xỏc nh trang 44) d2 d x x x N dx dx2 D = lim + N k2 k=1 bi cụng thc (xem B l toỏn t vi phõn Bessel (xem trang 75) (z) l hm Gamma, (z) = tz1 et dt, (z) > 0 KL l bin i tớch phõn Kontorovich-Lebedev (xem trang 8, 22, 23) K (z) l hm Macdonald (xem trang 20) L l toỏn t vi phõn bc hai c xỏc nh bi cụng thc L= Lp (R+ ), x + 3x + x2 x x p < , l khụng gian cỏc hm s f xỏc nh trờn R+ , tho f Lp (R+ ) p |f (x)| dx = p < Lp (R+ , ), R+ , tho p < , l khụng gian cỏc hm s f xỏc nh trờn trờn f Lp (R+ ,) p |f (x)| (x)dx = p < , õy (x) l mt hm trng dng L (R+ ) l khụng gian gm cỏc hm b chn theo chun ess sup trờn R+ f = ess sup |f | := inf{M > : à(x R+ : |f (x)| > M ) = 0, h.k.n.} (ã ã) l tớch chp Kontorovich-Lebedev (xem trang 9) (ã ã) l tớch chp Fourier (xem trang 10) (ã ã) l tớch chp suy rng Kontorovich-Lebedev - Fourier cosine th KL F nht (xem trang 25) (ã ã) l tớch chp suy rng Kontorovich-Lebedev - Fourier cosine th hai (xem trang 25) (ã ã) l tớch chp suy rng Kontorovich-Lebedev - Fourier sine (xem trang 25) (ã ã) l tớch chp suy rng Kontorovich-Lebedev - Fourier sine - Fourier cosine (xem trang 34) T1,h l bin i tớch phõn kiu tớch chp Kontorovich-Lebedev (xem trang 25) Th l bin i tớch phõn kiu tớch chp suy rng Kontorovich-LebedevFourier (xem trang 45) f (z) M < vi mi z thuc f (z) = O(g(z)), z a, cú ngha l g(z) vo mt lõn cn ca a f (z) f (z) = (g(z)), z a, cú ngha l lim = za g(z) f (z) f (z) g(z), z a, cú ngha l lim = za g(z) M U Tng quan v hng nghiờn cu v lý chn ti Bờn cnh nhng bin i tớch phõn ni ting cú vai trũ quan trng gii tớch toỏn hc núi riờng v cỏc ngnh khoa hc núi chung nh cỏc bin i tớch phõn Fourier, Laplace, Mellin, Hankel , nhng nm 38-39 ca th k trc, hai nh toỏn hc Nga l Kontorovich M.I v Lebedev N.N nghiờn cu bi toỏn v nhiu x súng in t vi biờn hỡnh nờm ó xõy dng bin i tớch phõn m sau ny c gi l bin i tớch phõn KontorovichLebedev (xem [29, 30, 67]) Cỏc tớnh cht ca bin i tớch phõn KontorovichLebedev khụng gian L1 , L2 , cụng thc bin i ngc v cỏc ng dng c nghiờn cu sau ú bi Lebedev N.N., Sneddon I.N., Lowndes J.S., Jones D.S (xem [24, 33, 34, 35, 36, 40, 41, 57]) nh ca hm f qua phộp bin i tớch phõn Kontorovich-Lebedev, kớ hiu l KL[f ], c xỏc nh bi cụng thc KL[f ](y) = Kiy (x)f (x)dx, y R+ , (0.1) vi K (x) l hm Macdonald cú ch s thun o = iy (xem [29, 30, 67]) iu ỏng chỳ ý, khỏc vi cỏc bin i tớch phõn k trờn, nhõn ca phộp bin i tớch phõn ny l hm c bit Macdonald, mt nhng hm cú nhiu ng dng khoa hc v k thut n nay, nhng kt qu v bin i tớch phõn Kontorovich-Lebedev trờn cỏc khụng gian hm vi h to tr, h to cu; khụng gian Lebesgue Lp vi trng cng nh xem xột trờn khụng gian hm suy rng ó khỏ phong phỳ v sõu sc (xem [18, 20, 21, 66, 71, 81]) Bin i tớch phõn KontorovichLebedev trờn khụng gian hai chiu, khụng gian nhiu chiu; bin i ri rc, bin i hu hn liờn quan n bin i tớch phõn Kontorovich-Lebedev cng ó c cỏc nh toỏn hc quan tõm nghiờn cu (xem [72, 78, 82]) Cựng vi cỏc bin i tớch phõn k trờn, tớch chp i vi cỏc bin i tớch phõn ny ó c xõy dng v ng dng nhiu lnh vc khỏc Nm 1998, Kakichev V.A v Thao N.X ó a nh ngha tớch chp suy rng f h vi hm trng ca hai hm f v h i vi ba phộp bin i (G(ã, t) (F ))(x) = K0 (x) L1 (R+ ;K0 ()) x p 1p |(x)|dx 1p L1 (R+ ;K0 ()) =()p1 p (g1 (ã, t) (F2 ) Lp (R+ ;xp K01p (x)) (4.57) Ngoi ra, x L1 (R+ ; ex ) = (4.58) Vỡ th, t (3.16) v (4.57) ta suy (g1 (ã, t)) (F ) Lp (R+ ;xp K01p (x)) p1 L1 (R+ ;K0 ()) g1 (ã, t) Lp (R+ ) Gi s u0 Lp (R+ ; (2 ())1p K0 ()), ta cú F2 () = F2 Lp (R+ ;2 ()K0 ()) (4.59) u0 () Lp (R+ ; ()K0 ()), () v u(x, t) Lp (R+ ;xp K01p (x)) = g1 (ã, t) (F2 ) 1 pL11 (R+ ;K0 ()) g1 (ã, t) = pL11 (R+ ;K0 ()) g1 (ã, t) T ú, ta nhn c Mnh Lp (R+ ;xp K01p (x)) Lp (R+ ) F2 Lp (R+ ;2 ()K0 ()) Lp (R+ ) u0 Lp (R+ ;(2 ())1p K0 ()) Mnh 4.3.1 Gi s v u0 l cỏc hm tho iu kin (C1)-(C2) Phng trỡnh dng parabolic (4.26) vi iu kin ban u (4.27) trng hp H(u, x, t, ) cú dng (4.30) cú nghim nht khụng gian hm SKL(R+ ) v cú cụng thc dng tớch chp Kontorovich-Lebedev - Fourier cosine th nht u(x, t) = (g1 (ã, t) f ))(x) (4.60) Nu thờm gi thit g1 (ã, t) Lp (R+ ), t R+ , u0 Lp (R+ ; ())1p K0 ()), p l s thc ln hn 1, (x) L1 (R+ ; K0 (v)), thỡ ta cú c lng ca nghim trờn khụng gian Lp (R+ ; xp K01p (x)) u(x, t) Lp (R+ ;xp K01p (x)) p1 L1 (R+ ;K0 ()) g1 (ã, t) 101 Lp (R+ ) u0 Lp (R+ ;(2 ())1p K0 ()) , (4.61) vi p1 l s m liờn hp ca p 4.3.2 Phng trỡnh parabolic phi tuyn liờn quan tớch chp Kontorovich-Lebedev Cui cựng, ta ng dng phng phỏp tớch chp Kontorovich-Lebedev tỡm nghim trng hp c bit ca phng trỡnh (4.26) H(u, x, t, ) = 2x x x e ( x + + ) u(, t)d, (4.62) vi u(x, t) thuc lp C 2,1 (R2+ ) v u(x, t) SKL(R+ ) Khi ú, phng trỡnh (4.26) cú dng phi tuyn, c vit li di dng u(x, t) = B[xu(x, t)] + (u(ã, t) u(ã, t))(x) KL t x (4.63) Ta xột iu kin ban u u(x, 0) = u0 (x), x R+ , (4.64) y2 p vi u0 L (R+ ) v tn ti s thc dng C cho KL[u0 ](y) 1+C dng bin i Kontorovich-Lebedev theo bin x c hai v ca phng trỡnh (4.63), ta nhn c Kiy (x)B[xu(x, t)]dx + [U (y, t)]2 , x Ut (y, t) = (4.65) õy U (y, t) := KL[u](y, t) l nh qua bin i Kontorovich-Lebedev theo bin x ca u(x, t) S dng cụng thc bin i ngc Kontorovich-Lebedev (xem [67]) u(x, t) = KL1 [U (, t)] := x Ki (x) sinh U (, t) d, 102 (4.66) ta cú KL B[xu(x, t)] (y, t) = x Kiy (x) x B[Ki (x)] sinh U (, t) d dx =KL KL [ U (, t)] (y, t) = y U (y, t) (4.67) H qu l chỳng ta cú th bin i phng trỡnh (4.65) v dng phng trỡnh Riccati phi tuyn Ut (y, t) = y U (y, t) + [U (y, t)]2 (4.68) vi iu kin ban u U (y, 0) = KL[u0 ](y), y R+ (4.69) Phng trỡnh (4.68) l phng trỡnh 1.2.2.24 c nghiờn cu [47] wt (y, t) ta bin i (4.68) v phng trỡnh vi Bng phộp th U (y, t) = w(y, t) phõn dng cụng thc 2.1.2.11 [47] wtt (y, t) + y wt (y, t) = 0, (4.70) m nghim ca nú cú dng w(y, t) = C1 (y)ey t + C2 (y) (4.71) Do ú, bng tớnh toỏn n gin ta nhn c nghim ca phng trỡnh trờn l y2 U (y, t) = , (4.72) + C(y)ey2 t (y) vi C(y) = CC12 (y) T iu kin ban u (4.69) i vi U (y, t) ta cú y2 U (y, 0) = KL[u0 ](y) = + C(y) Do ú, y2 C(y) = C > KL[u0 ](y) 103 y2 Ta nhn c U (y, t) y2 t , y, t > Vỡ vy, U (y, t) thuc khụng gian Ce y L2 (R+ ; (1 + y) e ), t R+ , v Ut (y, t) L1 (R+ ; (1 + y)1/2 ey/2 ), t R+ , tho iu kin (A1) v (A2) Do ú, u(x, t) c tớnh t cụng thc (4.66) l nghim nht ca phng trỡnh parabolic phi tuyn (4.63) trờn khụng gian SKL(R+ ) vi iu kin ban u (4.64) Bỡnh lun cui Chng Trng nhiu x súng õm v trng nhiu x súng in t vi tr khỏng nún trũn l mt nhng c bn lý thuyt nhiu x v luụn nhn c s quan tõm nghiờn cu ca cỏc nh khoa hc vi phng phỏp tip cn phong phỳ Trong nhng cụng trỡnh nghiờn cu gn õy, cỏc tỏc gi nh Lyalinov M.I., Zhu N.Y., Bernard J.M.L ó t c mt s kt qu quan trng v bi toỏn ny bng cỏch s dng biu din tớch phõn Kontorovich-Lebedev cho nhiu x trng súng õm, th Debye ca nhiu x trng súng in t thụng qua phng phỏp tỏch bin khụng hon ton Tuy nhiờn, v mt toỏn hc nhng tớnh cht v hm ph cng nh nhng c lng liờn quan n biờn ca trng nhiu x cng cha c nghiờn cu Vỡ vy, chỳng tụi hy vng biu din cỏc i lng vt lý quan trng ny qua tớch chp suy rng Kontorovich-Lebedev - Fourier, t ú cung cp thờm cỏc cụng c toỏn hc gii quyt cú hiu qu mt s liờn quan Trong chng ny, chỳng tụi ó cú nhng tip cn ban u v nhn c c lng nhiu x trng súng õm (4.4), c lng th Debye (4.19) Trong mt trng hp riờng, chỳng tụi ó nhn c mt s kt qu nh c lng cho trng nhiu x súng õm trờn cỏc khụng gian Lp vi trng hay xỏc nh th ca hm ph i vi th Debye dng c nhng tớnh cht liờn quan n tớch chp suy rng Kontorovich-Lebedev - Fourier c nghiờn cu Chng v Chng S dng bin i tớch phõn gii cỏc phng trỡnh o hm riờng l mt phng phỏp hiu qu, c bit l cỏc bin i tớch phõn Fourier, Laplace, Mellin õy, chỳng tụi s dng bin i tớch phõn Kontorovich-Lebedev v cỏc tớch chp suy rng Kontorovich-Lebedev - Fourier nghiờn cu mt lp cỏc phng trỡnh vi-tớch phõn o hm riờng v ó nhn c cụng thc nghim dng úng cng nh nhn c ỏnh giỏ tiờn nghim cho lp phng trỡnh ny 104 Kt lun Chng Cỏc kt qu chớnh t c: Nhn c c lng biờn ca trng nhiu x súng õm U (x) Biu din U (x) theo tớch chp suy rng Kontorovich-Lebedev - Fourier v nhn c c lng theo chun ca U (x) trờn cỏc khụng gian Lp vi trng Nhn c c lng biờn ca th Debye trng nhiu x súng in t usj , vjs Biu din usj , vjs theo tớch chp suy rng KontorovichLebedev - Fourier, tỡm c c lng a phng ca th Debye v xỏc nh hm ph vi d liu cho trc Xõy dng cụng thc nghim dng tớch chp, tớch chp suy rng Kontorovich-Lebedev ca mt lp phng trỡnh dng parabolic v ỏnh giỏ chun ca nghim trờn cỏc khụng gian Lebesgue 105 KT LUN Cỏc kt qu chớnh ca Lun ỏn l: Xõy dng tớch chp suy rng i vi cỏc phộp bin i tớch phõn Kontorovich-Lebedev, Fourier sine, Fourier cosine Nhn c tớnh cht toỏn t ca tớch chp suy rng, ng thc nhõn t húa, ng thc kiu Parseval, nh lý kiu Titchmarsh Nhn c iu kin cn v phộp bin i vi-tớch phõn kiu tớch chp suy rng Kontorovich-Lebedev - Fourier sine - Fourier cosine l ng cu, ng c gia hai khụng gian L2 (R+ ) v L2 (R+ ; x) Xõy dng bt ng thc kiu Young, kiu Saitoh, kiu Saitoh ngc trờn cỏc khụng gian Lp vi trng i vi cỏc tớch chp suy rng KontorovichLebedev - Fourier Nhn c ng dng gii v ỏnh giỏ nghim mt s lp phng trỡnh vi tớch phõn v phng trỡnh o hm riờng dng parabolic Thit lp biu din theo tớch chp suy rng Kontorovich-Lebedev - Fourier vi tr khỏng nún trũn ca nhiu x trng súng õm, th Debye ca trng nhiu x súng in t v nhn c cỏc c lng im, c lng theo chun ca cỏc i lng ny Mt s m cú th tip tc nghiờn cu: Nghiờn cu hng s tt nht vi cỏc bt ng thc ó nhn c i vi cỏc tớch chp suy rng Kontorovich-Lebedev - Fourier Nghiờn cu bin i tớch phõn Kontorovich-Lebedev v cỏc tớch chp suy rng i vi bin i tớch phõn ny mt s bi toỏn vt lý cú liờn quan Xõy dng v nghiờn cu tớch chp, tớch chp suy rng i vi bin i Kontorovich-Lebedev ri rc v Kontorovich-Lebedev hu hn v cỏc ng dng 106 DANH MC CễNG TRèNH CễNG B CA LUN N Thao N X., Tuan V K., Hoang P V., and Hong N.T (2016), Asymptotics of the scattered Debye potentials via a generalized convolution, Integral Transforms Spec Funct., Vol 27, No 2, 126-136 Tuan T., Hong N.T., and Hoang P.V (2016), Generalized convolution for the Kontorovich-Lebedev, Fourier transforms and applications to acoustic fields, Acta Math Viet., Vol 42, No 2, 355-367 Hong N.T., Hoang P.V., Tuan V.K 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Kontorovich-Lebedev, Fourier, Fourier sine, Fourier cosine v nhng nh lý, mnh cú liờn quan n Lun ỏn Chng xõy dng tớch chp suy rng mi i vi cỏc bin i tớch phõn Kontorovich-Lebedev, Fourier sine, Fourier cosine... ã) l tớch chp Fourier (xem trang 10) (ã ã) l tớch chp suy rng Kontorovich-Lebedev - Fourier cosine th KL F nht (xem trang 25) (ã ã) l tớch chp suy rng Kontorovich-Lebedev - Fourier cosine