格林函数

Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ‚�£Green¤¼ê{ úôŒÆêÆX ÅV= February 1, 2008 úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ‚�£Green¤¼ê{ ‚�£Green¤¼ê{´ ‡©§¦)ff˜«­‡{§§‚ ̇ïÄgŽ´r˜a>Ł¯Kff¦)=z,˜‡AÏ>Ł¯ Kff¦)(‚�¼êffÏé)§Œ±^ù‡AÏ>Ł¯Kff)5L «˜a>Ł¯Kff). úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn Greenúª,éX²¡k.«Sff�­È©†«>.­‚ þff­‚È©; Gaussúª, éX˜mk.«Sffn­È©†«>.4­ ¡þ­¡È©; 3Rn˜m¥ffk.«´Äkaqff(J? úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn Greenúª,éX²¡k.«Sff�­È©†«>.­‚ þff­‚È©; Gaussúª, éX˜mk.«Sffn­È©†«>.4­ ¡þ­¡È©; 3Rn˜m¥ffk.«´Äkaqff(J? úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn Greenúª,éX²¡k.«Sff�­È©†«>.­‚ þff­‚È©; Gaussúª, éX˜mk.«Sffn­È©†«>.4­ ¡þ­¡È©; 3Rn˜m¥ffk.«´Äkaqff(J? úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn—n = 2žGreenúª ²¡k.«Ω ⊂ R2,>.4­‚∂Ω, P(x , y)ÚQ(x , y)´ëY  �êff��¼ê,KGreenúª´∫ ∫ Ω ( ∂Q ∂x − ∂P ∂y ) dxdy = ∫ ∂Ω P(x , y)dx + Q(x , y)dy P∂Ωffü ƒ•þ(_ž�•) −→ T = (α, β),ü  {• þ −→n = (β,−α)(„ã). � � � úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn—n = 2žGreenúª ddx = αds,dy = βds, Ù¥dsL«∂Ωffl�ƒ,∫ ∫ Ω ( ∂Q ∂x − ∂P ∂y ) dxdy = ∫ ∂Ω (P(x , y)α+ Q(x , y)β)ds = ∫ ∂Ω (Q,−P) · −→n ds. AO,eP−→w = (Q,−P),K∫ ∫ Ω div−→w dxdy = ∫ ∂Ω −→w · −→n ds. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn—n = 3žGaussúª k.«Ω ⊂ R3,>.4­¡∂Ω, P(x , y , z),Q(x , y , z)ÚR(x , y , z)k˜ffiëY �ê, KGaussúª ´∫ ∫ ∫ Ω ( ∂P ∂x + ∂Q ∂y + ∂R ∂z ) dxdydz = ∫ ∫ ∂Ω (Pα+ Qβ + Rγ)ds Ù¥ −→n = (α, β, γ)´>.­¡∂Ωffü  {•þ. P −→w = (P,Q,R),KGaussúªŒ±�¤∫ ∫ ∫ Ω div−→w dxdydz = ∫ ∫ ∂Ω −→w · −→n ds úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Ñݽn GreenúªÚGaussúªkƒÓff/ª,r§í2�˜„/ª: ÑÑÑÝÝݽ½½nnn:Ω´Rn¥ffk.«,>.∂Ω, −→n´>.∂Ωþffü  {•þ, −→w´ΩSffäk˜ffiëY �êff•þ¼ê. K∫ Ω div−→w dV = ∫ ∂Ω −→w · −→n dS . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÑÑÑÝÝݽ½½nnn Greenúúúªªª Greenúª Ω´Rn¥ffk.«, >.∂Ω,−→n´>.þffü  {•þ, uÚv3Ωk�ffiëYff �ê. �−→w = u∇v , | ^div−→w = u4v +∇u∇v , ·‚k(1˜Greenúª):∫ Ω u4vdV = ∫ ∂Ω u ∂v ∂−→n dS − ∫ Ω ∇u · ∇vdV . (1) �†¼êu†vaqŒ�∫ Ω v4udV = ∫ ∂Ω v ∂u ∂−→n dS − ∫ Ω ∇u · ∇vdV . (2) (1)Ú(2)ƒ~Œ�(1�Greenúª):∫ Ω (v4u − u4v)dV = ∫ ∂Ω ( v ∂u ∂−→n − u ∂v ∂−→n ) dS . (3) úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÄÄÄflflfl))) Green¼¼¼êêê½½½ÂÂÂ999555ŸŸŸ Äfl)ff½Â ½½½ÂÂÂ: L´˜‡‚5‡©Žf, M0´Rn¥ff˜‡ff½:,M´Rn¥ ff:. r§Lu = δ(M −M0)ff)U(M,M0)¡ §Lu = f (M)½Lu = 0ffÄfl). úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÄÄÄflflfl))) Green¼¼¼êêê½½½ÂÂÂ999555ŸŸŸ Äfl)ff½Â ~~~. ¦n‘Laplace§ffÄfl). ))).�M0 = (x0, y0, z0), M = (x , y , z),Äfl)U(M,M0)§ −4U = − ( ∂2U ∂x2 + ∂2U ∂y2 + ∂2U ∂z2 ) = δ(x − x0, y − y0, z − z0) (4) ff). Ú?±M0 = (x0, y0, z0)¥%ff4‹I: x = x0 + r cos θ cosϕ, y = y0 + r cos θ sinϕ, z = z0 + r sin θ, ¦§(4)ff==†rk'ff)U(r)(†θÚϕÃ')÷v −4U = − 1 r2 d dr ( r2 d2U dr2 ) = δ(r), úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÄÄÄflflfl))) Green¼¼¼êêê½½½ÂÂÂ999555ŸŸŸ Äfl)ff½Â �r > 0ž,d− 1 r2 d dr ( r2 d 2U dr2 ) = 0�U(r) = A+ Br ,Ù¥AÚB´? ¿~ê. ·‚e¡·�À�AÚB¦�§3r = 0:„U÷v§. 3 ¥D� = {M : |M −M0| < �}þ^1�Greenúª(u=U, v=1) − ∫ ∫ ∫ D� 4UdV = ∫ ∫ ∫ D� δ(M −M0)dV = 1, − ∫ ∫ ∫ D� 4UdS = − ∫ ∫ S� ∂U ∂−→n dS = − ∫ ∫ S� dU dr dS = − ∫ ∫ S� B r2 dS = 4piB, ÏdB = 14pi .{üPA = 0,Äfl)´U = 1 4pir ,= U(M,M0) = 1 4pi √ (x − x0)2 + (y − y0)2 + (z − z0)2 . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÄÄÄflflfl))) Green¼¼¼êêê½½½ÂÂÂ999555ŸŸŸ Äfl)ff½Â aq/,·‚Œ±���‘Laplace§ffÄfl) U(M,M0) = − 1 2pi ln √ (x − x0)2 + (y − y0)2. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÄÄÄflflfl))) Green¼¼¼êêê½½½ÂÂÂ999555ŸŸŸ Green¼êff½Â ½½½ÂÂÂ:�Ω´Rn¥ff˜‡k.«, >.S , M0 ∈ Ω. ¡÷v −4MG = δ(M −M0), M ∈ Ω; G = 0, M ∈ S ff)G (M,M0)´Laplace§÷vDirichlet>Ł^‡ffGreen¼ê. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ ÄÄÄflflfl))) Green¼¼¼êêê½½½ÂÂÂ999555ŸŸŸ Green¼êff5Ÿ 555ŸŸŸ: Green¼êäké¡5, =G (M1,M2) = G (M2,M1). yyy²²² 5¿� −4MG (M,M1) = δ(M −M1), M ∈ Ω; G (M,M1) = 0M ∈ S , −4MG (M,M2) = δ(M −M2), M ∈ Ω; G (M,M2) = 0M ∈ S . �u(M) = G (M,M1), v(M) = G (M,M2). 1�Greenúª� 0 = ∫ S [ G (M,M1) ∂G (M,M2) ∂−→n − G (M,M2) ∂G (M,M1) ∂−→n ] dS = ∫ Ω [G (M,M1)4MG (M,M2)− G (M,M2)4MG (M,M1)] dV = ∫ Ω [G (M,M1)δ(M,M2)− G (M,M2)δ(M,M1)] dV = G (M2,M1)− G (M1,M2). úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Green¼ê{—Poisson§ffDirichlet¯K ½½½nnn:�Ω´Rn¥ff˜‡k.«, >.S , M0 ∈ Ω, G (M,M0)´Laplace§÷vDirichlet>Ł^‡ffGreen¼ê,K −4u = f (M), M ∈ Ω; u = ϕ(M), M ∈ S ff)ŒL« u(M0) = ∫ S ϕ(M) ∂G (M,M0) ∂−→n dSM + ∫ Ω G (M,M0)f (M)dVM . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Green¼ê{—Poisson§ffDirichlet¯K yyy²²². éuM0 ∈ Ω,d1�Greenúª u(M0) = ∫ Ω δ(M −M0)u(M)dVM = − ∫ Ω u(M)4MG (M,M0)dVM = − ∫ Ω G (M,M0)4Mu(M)dVM − ∫ S [ u(M) ∂G (M,M0) ∂−→n − G (M,M0) ∂u(M) ∂−→n ] dSM = ∫ Ω G (M,M0)f (M)dVM − ∫ S ϕ(M) ∂G (M,M0) ∂−→n dSM úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê XÛ¦Green¼ê? �Ω´Rn¥ff˜‡k.«, >.S ,?ØPoisson§ ffDirichlet¯K −4u = f (M), M ∈ Ω; u = ϕ(M), M ∈ S . ·‚‡¦ÑLaplace§ffGreen¼ê, = −4MG (M,M0) = δ(M−M0), M ∈ Ω; G (M,M0) = ϕ(M), M ∈ S ff). PU(M,M0)´Laplace§ffÄfl), (�n = 3žU(M,M0) = 1 4pirMM0 , �n = 2žU(M,M0) = − 12pi ln rMM0) u(M) = G (M,M0)− U(M,M0),K −4u = 0, M ∈ Ω; u = U(M,M0), M ∈ S . Ïd,·‚‡é˜‡3>.S�ŁU(M,M0)ffNÚ¼êu(M), G (M,M0) = u(M) + U(M,M0). úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Green¼êffÔn¿Â XJ·‚3M0:˜˜˜‡ü �>Ö,@o3>.SffSýka AffK>Ö,3>.Sff ýkaAff�>Ö.XJ·‚rΩ� /,@o3>.SþkkaAffK>Ö.3M0:ffü �>ÖÚ åff> U(M,M0),aAffK>Ö�)ff> u(M), Ï d,Green¼êG (M,M0)´dM0:ffü �>ÖÚ§ffaAK> Ö�)�Ó3M:¤�)ff> rÝ(„ã). � � úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Œ˜mþffGreen¼ê Ω = {(x , y , z) : z > 0}, S = {(x , y , z) : z = 0}. 3M0(x0, y0, z0)˜˜˜‡ü �>Ö,�M0'uxoy²¡ffé¡ :M1(x0, y0,−z0)…3M1:˜˜˜‡ü K>Ö,§‚�)ff>  3xoy²¡þƒp-ž,=M1:ffü K>Ö¤åffŁ^†M0: ü �>ÖffaA>Ö¤åffŁ^˜�(„ã) � � � � � � � úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Œ˜mþffGreen¼ê M0:ü �>ÖÚåff> rݏ 1 4pirMM0 ,aA>ÖÚå(=M1ü  K>Ö¤å)ff> rݏ− 14pirMM1 ,l þŒ²¡ffGreen¼ê  G (M,M0) = 1 4pirMM0 − 1 4pirMM1 = 1 4pi √ (x − x0)2 + (y − y0)2 + (z − z0)2 − 1 4pi √ (x − x0)2 + (y − y0)2 + (z + z0)2 . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Œ˜mþffGreen¼ê ~~~.¦)−4u = f (x , y , z), z > 0; u(x , y , 0) = ϕ(x , y). ))).éu«Ω = {(x , y , z) : z > 0}, k−→n = (0, 0,−1),l ∂G ∂−→n = − ∂G ∂z = − z0 2pi[(x − x0)2 + (y − y0)2 + z20 ]3/2 . u(x0, y0, z0) = ∫ Ω G (M,M0)f (M)dVM − ∫ S ϕ(M) ∂G (M,M0) ∂−→n dSM = 1 4pi ∫ z>0 ( 1√ (x − x0)2 + (y − y0)2 + (z − z0)2 − 1√ (x − x0)2 + (y − y0)2 + (z + z0)2 ) f (x , y , z)dxdydz + 1 2pi ∫ (x ,y)∈R2 z0ϕ(x , y) [(x − x0)2 + (y − y0)2 + z20 ]3/2 dxdy . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê ¥þffGreen¼ê Ω = {(x , y , z) : x2 + y2 + z2 < R2}, S = {(x , y , z) : x2 + y2 + z2 = R2}. �M0(x0, y0, z0) ∈ Ω, ë �OM0¿ò�OM1¦�OM0 · OM1 = R2(¡M1ÚM0´'uŒ »Rff¥¡é¡:)(„ã). � � � � � � éu¥¡Sþff?¿:P ∈ S ,4OM1P v 4OM0P, l PM0 PM1 = OM0 R , ∀P ∈ S . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê ¥þffGreen¼ê e·‚3M0(x0, y0, z0)˜˜˜‡ü �>Ö,3M1:˜˜>þ −qffK>Ö,K§‚3p ∈ S?�)ff> rݏ 1 4pi [ 1 rPM0 − q rPM1 ] , AO,�·‚�q = RM0 (§´˜‡~ê)ž,é?¿ffP ∈ S§‚�) ff> rݏ",=M1:ffK>Ö¤åffŁ^†M0:ü �>Ö ffaA>Ö¤åffŁ^˜�.l ·‚��¥þffGreen¼ê G (M,M0) = 1 4pi [ 1 rMM0 − R rOM0 1 rMM1 ] . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê ¥þffGreen¼ê eM0 = (x0, y0, z0),@o M1(x1, y1, z1) = rOM1 rOM0 = ( x0R x20 + y 2 0 + z 2 0 , y0R x20 + y 2 0 + z 2 0 , z0R x20 + y 2 0 + z 2 0 ) , G (M,M0) = 1 4pi [ 1√ (x − x0)2 + (y − y0)2 + (z − z0)2 − R√ x20 + y 2 0 + z 2 0 1√ (x − x1)2 + (y − y1)2 + (z − z1)2  . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Greenúúúªªª Green¼¼¼êêê Green¼¼¼êêê{{{ Poisson§§§ffffffDirichlet¯¯¯KKK AAAÏÏÏ«««þþþffffffGreen¼¼¼êêê Ïd§ u(x , y) = (1 + x)y + 1, x ≥ 0, y ≥ 0. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== ‚‚‚���£££Green¤¤¤¼¼¼êêê{{{ Green公式 散度定理 Green公式 Green函数 基本解 Green函数定义及性质 Green函数法 Poisson方程的Dirichlet问题 特殊区域上的Green函数