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Chuong 1. PhiJOng trinh e3ao ham rieng: Dinh nghia va vi dti 11
1.1. Su XLiai hii}n ciia phirong tr'inh elao ham rieng (DHR) va moi quan
he cua chi'ing voi cac linh vuc khae 11
1 ,2. Cac dinh nghia chung \'e phirong trinh DHR 13
I .3. Cac VI du lieu bicu 15
1.3.1. Cac phirong trinh DHR 15
1.3.2, lie phuong trinh DHR 18
1 .4, Nhung di6u can chi'i y khi nghien ciru phuong trinh DHR 19
1.4.1, Bai loan dat chinh, Nghicim ce3 dicii 19
1.4.2. NghiiJm ycii. Tfnh ehi'nh quy 19
C h u o n g 2. Ky hieu va kien thttc phu trd 21
2 1, Ky hmi 21
2.1.1. Ky hicu de"ii \'ci'i ni;i uan 21
2.1.2. Ky hicii hinh hoe 22
2.1.3. Ky hi(5u c;ic ham so 23
2.1.4. Ky hieu dao ham 24
2.1.5. Cac khong gian ham 26
2.1.6. Ham vc;c to 26
2.1.7. Ky hieu e;ic u6c lui7ng 27
2.1.8. Mot so quy uac vc ky hieu 28
2.2. Bai dang ihirc 28
2.2.1. Ham loi 28 1.2.2. Mot so Bai dang thiic co ban 2"
2.3. Mot so kien ihuc ve giai ti'ch thirc 33
2.3.1. Bien 33
2.3.2. Dinh ly Gauss-Green 3.''
2.3.3. Tga do cue, ceiing thiic do'i mi^n 3('
2.3.4. Tich chap va dp iron 37
2.3.5. Dinh ly ham nguoc 40
2.3.6. Dinh ly ham an 40
2.3.7. Hoi tu dSu 41
2.4. Mot so kieii ihiic \'e giai li'ch ham 4?
2.4.!.-. Khong gian Banach 42
2.4.2. Khong gian Hilbert 43
2.4.3. loan lu tuyeSii ti'nh hi chan 44
2.4.4. Hgi tu yeiu 46
2.5. Vc ly thuyei do do 47
2.5.1. F>o do Lehesgue 4
2.5.2. Ham do duc;e va u'eh phan 48
2.5.3. C'dc dinh ly hoi tu dcii vcii ti'ch phSn 49
2.5.4. Phep loan vi phan 50
2.5.5. Ham nh:ni gia tri irong khong gian Banach 50
ChiiOng 3. NhxJng phtfcfng trinh dao ham rieng tuyen tinh quan
tr9ng 53
3.1. Phuerng trinh chuyi5n dich 54
3.1.1. Bai loan gia tri ban dau 54
3.1.2. Bai loan khong thuan nhat 55
3.2. Phuong trinh Laplace 55
3.2.1. Nghiem co ban SI
3.2.2. Cong thiic gia tri chinh 61 3.2.3, Cac li'iih chat cua ham dic^Hi hoa 63
3.2.4, Ham Green , , ., 69
3.2.5, Phirong ph;ip nang luong 77
3.3. Phucmg trinh truyein nhiet 79
3.3.1. NghieMU CO ban 80
3.3.2. Cong thiie gia tri chinh 87
3.3.3. Mot s(i tinh chai cua nghiem 91
3.3.4. Phuong ph;ip nang luong 99
3.4. Phuong trinh truyel-n scing 101
3.4.1, Nghiem theo irung binh ciiu 103
3.4.2, Bai loan khong thuan nhai 116
3.4.3, Phuwng phiip nang hrgng 118
3.5. Bai tap Chucmg 3 120
Clifofng 4. Phifong trinh dao ham rieng phi tuyen cap mot 125
4.1. Ti'ch phan day dii. hinh bao 126
4.1.1. Tich phan day dii 126
4.1.2. Nhung nghiem mc)i Iir hinh bao 128
4.2. Dae trirng 131
4.2.1. Phutmg uinh vi phan thiivmg dac trung 131
4.2.2. Cac \i du 133
4.2.3. Ditni kien bie;n 137
4.2.4. Nghiem dia phuc^ng 140
4.2.5. Uiig dung 144
4..3. Phuong trinh Hamilton-Jacobi 150
4.3.1. Phep tinh bien phan, phuong trinh vi phan thuong Hamilton 150
4.3.2. Bien ddi Legendre, cong thiic Hopf-Lax 155
4.3.3. Nghiem yeu, tinh duy nhat 163
i.4. Dinh luat bao loan 170 4.4.1. Soc, di(}u kien entropi 17i)
4.4.2. Cong thiic Dax-Oleinik 17S
4.4.3. Nghie}m Kntropi. ti'nh duy nhai 18'
4.4.4. Bai locin Riemann i8S
4.4.5. Dang dicu nghiem khi f -^ oo 191
4.5. Bai tap Chirong 4 i9()
Chifdng 5. Mot so philo'ng p h a p b i e u d i e n n g h i e m 19'>
5.1. Phuong phap tach biei'n 19'*
5.2. Nghiem d6ng dang 20 i
5.2.1, .Song phang \a sc')ng Ian truydn, .Soliton .''.O '
5.2.2, Dong dang theo ty K- 211
5.3. Cac phuong phap bii:n dcii ti'ch phan 21.'
5.3.1, Biei'n deii Fourier 213
5.3.2, Bie5n e\6i Laplace 22,!
5.4. Bien deii phuong trinh phi tuyen tlianh tuyeii ti'nh 225
5.4.1, Biein de^l Hopf-Cole 225
5.4.2, Ham the vi 227
5.4.3, Bieii de')i Ic-ie do \-a bici'n de:)i l.egendre 22S
5.5. Phuc:(ng phap Laplace \'a thuan nhai hoa 231
5.5.1. Phuong phap Laplace 231
5.5.2. 'Hiuan nhai hoa 233
5.6. Chu6i luy thira 23(>
5.6.1. Mat khcing dac irung 23()
5.6.2. Ham giai ti'ch ihue 24J
5.6.3. Dinh ly Cauchy-Kovalevskaya 243
5.7. Bai tap Chuong 5 248
Tai l i e u tham k h a o 250
D a n h m u c tii khoa 253 CTion sach nay nam trong 136 sach Cao hoc do Vien Toan hoc chii tri va Nha
v;uat ban Dai hoc Quoe gia Ha Ngi an hanh. No dugc bien soan chii yeu dua vao
chuong trinh cao hoc eiia cac truong dai hge 6 My ve ehuyen nganh Phuong trinh vi
phan dao ham rieng trong nhiJng nam gan day va dugc GS L. Evans diic ket thanh
.sach "Partial Differential Equations". American Mathematical Society, 1998. Tac
gi;i CO gang lira chgn nhung kien thiic co ban nhat cua ly thuyei phuong trinh vi
phan dao ham rieng nham eung cap cho ban dgc mot each tiep can co hifeu qua den
linh vuc toan hoc co nhidu I'mg dung nay.
Khi bien soan tap tai lieu nay chiing toi da dua vao nhiJng phuong cham sau
day.
1. Chung t6i chii trgng den phuong trinh phi tuyen, vi noi chung ta thuong gap
chiing trong nhOng ung dung thirc te. Hon niia, mac du da dugc dS cap den
tir lau (trong the ky 18, 19), nhung ly thuyei cac phuerng trinh phi tuyen co
ban cho den ngay nay van chua dugc hoan chinh.
2. Mot bai loan phuctng trinh vi phan dao ham rieng, neu no co y nghia thuc
ti^n, thi chac chan no co nghiem, chi co dieu la nghiem do dugc hieu theo
nghia nao ma thoi. Nhifiu phuong trinh vi phan dao ham rifeng ma ta nghien
cuu. dac biet la phuong trinh phi tuyen dSu khong co nghiem c6 dien, vi vay
ta CO gang xay dimg ly thuyei cac nghiem suy rong hoac nghiem yeu cua
chiing, va di6u quan trgng 6' day la tinh duy nhat nghiem (do nhu cau ling
dung thuc te).
J. Khac veil rne^t so cucin sach khac thuong xSy dung ly thuyei phuong trinh vi
phan dao ham rieng theo each phan loai phuong trinh, 6' day chung tGi chii
trgng d6'n cac phuong phap iighien cUii thOng qua nhung vi du dac trung.
Cach lam nay nhkm cung ca'p cho ban dgc nhifeu phuong phap giai phuong
trinh vi phan dao ham rieng, di ho co the ap dung vao viec xem xet nhtrng
phuong trinh cii the trcmg thuc te. Ngoai ra, chiing toi quan tam dac biet den
viec tim nghiem ehinh xac ciia cac bai toan phuong trinh vi phan dao ham
rieng. xem do la mot trong nhiJng nhiem vu chinh ciia tap tai lieu nay.
Cu6'n sach dugc chia lam 2 tap. Trong tap 1, chiing toi se nhac lai nhiJng ki6'n
thirc chii yeu ve cac phuong trinh tuyen tinh quan trgng nhu: phuong trinh ehuyen
dich, phuong trinh Laplace, phuong trinh truy6n nhiet, phuong trinh truyen song,
/file/d/1cSgHMR ... sp=sharing
