Gauss div E = 1 ε 0 ρ(x, y, z), (1.3) E (x, y, z)ε 0 ρ(x, y, z) E = 0 (curl E = 0), (1.4) E = u(x, y, z), (1.5) u ( )(1.5) (1.3) u(x, y, z) = 1 ε 0 ρ(
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- 余 董
- 5 years ago
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1 Laplace Laplace() Poisson () Laplace Poisson Laplace Poisson ( ) GreenLaplace Green ( )Laplace Poisson Harnack Laplace Laplace 1. Laplace ( ) u n i=1 u x i = 0 (1.1) Poisson u n i=1 u x i = f(x 1,, x n ) (1.) u = u(x 1,, x n ) f(x 1,, x n ) n =, Laplace Poisson 1
2 Gauss div E = 1 ε 0 ρ(x, y, z), (1.3) E (x, y, z)ε 0 ρ(x, y, z) E = 0 (curl E = 0), (1.4) E = u(x, y, z), (1.5) u ( )(1.5) (1.3) u(x, y, z) = 1 ε 0 ρ(x, y, z). (1.6) (1.6) Poisson ρ 0 Laplace u = 0. (1.7) Laplace P 0 : (x 0, y 0, z 0 )mp : (x, y, z) F (x, y, z) m r P P 0 P 0 r = (x x 0 ) + (y y 0 ) + (z z 0 ) P 0 P F (x, y, z) ( m x x0 F (x, y, z) =, y y 0, z z ) 0. (1.8) r r r r F (1.8) F u(x, y, z) = m r (1.9) F = u.
3 ρ(x, y, z) dξdηdζ u(x, y, z) = ρ(ξ, η, ζ) (x ξ) + (y η) + (z ζ). (1.10) ulaplace u = u xx + u yy + u zz = 0, (1.11) ρ(x, y, z)hölder upoisson u = 4πρ. (1.1) (1.11)(1.1)u 0, (x, y, z) R\, u = 4πρ, (x, y, z) ρ(x, y, z) Hölder (1.13) ρ u t = T ( ) u x + u + F (t, x, y). (1.14) y F (x, y) u t (1.14) (1.15)Poisson u x + u (x, y) = F. (1.15) y T PoissonLaplace Laplace Laplace(1.1) Poisson(1.) 3
4 1.1 Laplace(1.1) (1.1) (1.1) (1.) ( ) t (1.1) (1.) ( Dirichlet ) R n g u = u(x 1,, x n ) (1.1)( (1.)) (1.16) ( Neumann) u = g. (1.16) Dirichlet Γ gu = u(x 1,, x n )Γ (1.1)( (1.)) Γ Γ n u n u n = g. (1.17) Γ (1.17) Neumann Dirichlet Neumann ( )u Laplace(1.1) (n = 3) u = g g 4
5 u = ϕ u ϕ ϕ Laplace(1.1) (n = 3) ϕ n = 0. Laplace Laplace Dirichlet Neumann LaplaceLaplace ΓΓ Dirichlet u Γ = 1. u 1 1 u = (x + y + z ) 1/ Laplace lim r u(x, y, z) = 0 (r = x + y + z ). (1.18) (1.18) u 1 1. Laplace Dirichlet R 3 Γ g u(x, y, z)(i)γ Q Laplace (ii) Q Γ(iii) (1.18) (iv) Γ Neumann u Γ = g. (1.19) Γ gu(x, y, z) (i) Γ Q Laplace (ii) Q Γ(iii) (1.18) (iv) ΓQ ñ(γ ) u ñ u ñ = g. (1.0) Γ 5
6 1. DirichletNeumann nlaplace 1.5 Poisson(1.)Poisson Laplace(1.1) fhölder (n = 3 (1.10) )Laplace(1.1) 1. u(x 1,, x n ) = f(r)( r = x x n) n ( u x1 x u xnx n = 0) f(r) = c 1 + c r n (n ), f(r) = c 1 + c ln 1 r (n = ), c 1, c. (r, θ, ϕ) u = 1 ( r u ) r r r 3. (r, θ, z) u = 1 r + 1 r sin θ r 4. (1) ax + by + c (a, b, c ); () x y xy; (3) x 3 3xy 3x y y 3 ; ( sin θ u ) + θ θ ( r u ) + 1 u r r θ + u z. 1 u r sin θ ϕ. (4) shny sin nx shny cos nx chny sin nx chny cos nx (n ); 6
7 shx (5) chx + cos y sin y chx + cos y. 5. Laplace (1) ln r θ; () r n cos nθ r n sin nθ(n ) (3) r ln r cos θ rθ sin θ r ln r sin θ + rθ cos θ. 6. Laplace (0 x a, 0 y b) u xx + u yy = 0, u(0, y) = u(a, y) = 0, u(x, 0) = sin πx, u(x, b) = 0. a 7. Laplace Dirichletu(x, y) 7
8 . ( ) Laplace Poisson Laplace Poisson Laplace( Poisson) ( ) ( ).1 Euler-Lagrange Hamilton q i (t)(i = 1,, n) q i (t)(i = 1,, n) Lagrange L (t, q i, q i ) [t 1, t ] t () δ t t 1 L (t, q i, q i )dt = 0. (.1) δq i t=t1 = 0, δq i t=t = 0 (i = 1,, n) (.1) (.) Lagrange L d ( ) L = 0 q i dt q i (i = 1,, n). (.) (.1) y = f(x)x x f( ) f (x) = 0( 1
9 f (x)( )f(x)( ) ) y = y(x)[a, b] () I[y] b a F (x, y(x), y (x))dx, (.3) F (.3)F y = y(x) I[y]y(x) y(x) I[y] y(x) F (x, y, y ) y(x) x y(a) = y, y(b) = y (.4) y y (.4)I[y] y(x)y(x) δy(x)y(x) + δy(x)δy(x) y(x) (.3) F (x, y + δy, y + δy ) = F (x, y, y ) + F F δy + y y δy, I[y + δy] I[y] = b a ( ) F F δy + y y δy (x)dx. I[y]δI[y] b ( ) F F δi[y] = δy + y y δy (x)dx. (.5).1 a I[y] F y d ( ) F = 0. dx y (.6)
10 y = y(x)i[y]y(x) δy(x) = εη(x), (.7) εη(x) η(a) = η(b) = 0. (.8) (.8)y + εη(.4).1. y y y(x) y 0 y(x) + εη(x) a b x.1: y(x) y(x) + εη(x) I[y + εη(x)] = b a F (x, y(x) + εη(x), y (x) + εη (x))dx (.9) εε = 0y(x) + εη(x) = y(x) y(x)i[y] ε = 0ε I[y + εη(x)] (.9) b a di[y + εη] dε = 0. ε=0 (.8)(.10) b ( ) F 0 = a y η (x)dx + F x=b y η x=a b [ F = y d ( )] F (x)η(x)dx. dx y a ( F y η + F ) y η (x)dx = 0. (.10) 3 b a [ d ( F dx y ) ] η (x)dx
11 η(x) F y d ( ) F = 0. dx y.1. (.6) Euler-Lagrange E-L n u(x 1,, x n ) I[u] = F (x 1,, x n, u(x 1,, x n ), u x1 (x 1,, x n ),, u xn (x 1,, x n )) dx 1 dx n, (.11) R n u u = g(x 1,, x n ), (x 1,, x n ). (.1).1. I[u] F n u i=1 x i ( F u xi ) = 0. (.13) u = u(x 1,, x n )I[u] u(x) u(x 1,, x n )δu(x)u(x) + δu(x) δu(x) u(x) δu(x) = εη(x) ε η(x) = η(x 1,, x n ) η = 0. (.14) (.14) u + εη(.1) (.11) [ F u η + n i=1 di[u + εη] dε = 0. ε=0 ] F η xi (x 1,, x n )dx 1 dx n = 0. (.15) u xi 4
12 Green(.14) (.15) F 0 = u ηdx 1 dx n + η ( Du F n) ds [ ] F n = η u ( ) Fuxi dx 1 dx n, x i i=1 η n i=1 x i ( Fuxi ) dx1 dx n (.16) η = η(x) (.14) Du F = (F ux1,, F uxn ) n η (.16) F n u (.13) i=1 x i ( F u xi (.13)(.11) Euler-Lagrange F F = 1 i=1 n ( ) u, x i E-L Laplace n u u = = 0. x i i=1 ) = 0. (.17) E-L.1 I[u 1,, u m ] = F (x, u 1 (x),, u m (x), Du 1 (x),, Du m (x))dx 1 dx n, ( ui x = (x 1,, x n ) Du i =,, u ) i x 1 x Euler-Lagrange n u u (.3) y(a) = y 0, y(b), (.18) 5
13 y 0 x = b y (.3) y = y(x) x = b I[y] x = b y(b) y = y(x) I[y](.3) F (x, y(x), y (x)) E-L(.16) δy(x) = εη(x) ε η(x) (.16) F y η(x) b a + b a η(a) = 0. (.19) di[y + εη] dε = 0 ε=0 [ F y d ( F dx y )] η(x)dx = 0. (.0) (.19) E-L(.16)η(b) (.0) F y = 0. (.1) x=b (.1) b y(b) y (b) I[u] = F (x, y, u(x, y), u x (x, y), u y (x, y))dxdy + G(s, u, u s )ds, (.) R s u s = u s F G..3 I[u]( (.) ) F u ( ) F ( ) F = 0 (.3) x u x y u y [ F dy u x ds F dx u y ds + G u d ( )] G = 0. ds u s (.4) 6
14 .3. δu(x, y) (.4) (.) F G F = u x + u y, G = σu, (.5) σ u E-L u = u xx + u yy = 0, (.6) ( dy u x ds u dx σu) y ds + = 0, ( σu) u n + = 0, (.7) n (.7) u (.6) (.7) Laplace. Oxy g(x, y)u(x, y) (x, y) ( Oxy) f(x, y) 7
15 V = =, { T [ ( ) ( ) ] } u u + x y fu dxdy, T ((6.7) )J(u) J(u) J(u) = { [ ( ) ( ) ] } 1 u u + fu dxdy, (.8) x y f = f T. ( g(x, y)) g V 0 = { v v C () C 1 (), v = g }, (.9) V 0 u u V 0 J(u) = min v V 0 {J(v)}. (.30) Laplace Poisson (.30) u = f,.1 (Dirichlet) u = g, (.31) (.30)u V 0 (.31) u (.31)V 0 u (.30) u (.30)(.11) F F = 1 [ ( ) ( ) ] u u + fu, x y 8
16 (.13)(.8)E-L u + f = 0. u V 0 V 0 ( (.9) ) u = g. u (.31) u V 0 Poisson (.31) w C () C 1 () w = 0 ( u + f)wdxdy = 0. Green( ) ( u w x x + u ) w y y fw dxdy = 0. (.3) v V 0 w = v u. w C () C 1 () w = 0, (.33) w (.3)J(v) (.3) ( u w J(v) = J(u + w) = J(u) + x x + u ) w y y fw dxdy + 1 [ ( ) ( ) ] w w + dxdy x y = J(u) + 1 [ ( ) ( ) ] w w + dxdy. x y (.33) J(v) J(u), w 0u(.30) u (.30).1 (.1) J(v) 9
17 J(v) V 0 Poisson (.31), (.8).3 E-L ( ) E-L ( + k )u = f ( ), (.34) ( σu) u n + = g, (.35) R n kf g σ n Ĩ[u] u { ( + k )u + f } dx. (.36) Green( )u, v C () C 1 () Ĩ[u] = (.35) Ĩ[u] = guds u vdx = v u n ds v udx, (.37) u u n ds u dx + k u dx + fudx. (.38) σu ds u dx + k u dx + fudx. 10
18 (.37) δĩ[u] = [( + k u )u + f]δudx n δuds σuδuds + gδuds = [( + k )u + f]δudx gδuds, (.35) (.34) (.36) Ĩ[u] I[u] Ĩ[u] + guds = u{( + k )u + f}dx + guds. (.34)-(.35)(.35) (.39) Rayleigh-Ritz (.39) Rayleigh-Ritz I[y] b a F (x, y(x), y (x))dx (.40) y(a) = y, y(b) = y, (.41) y y n c 1, c,, c n (.41) ỹ(x) = ϕ(x; c 1,, c n ), (.4) y(x)(.40)i[ỹ]c 1,, c n I[c 1,, c n ]. I[c 1,, c n ] c i = 0 (i = 1,,, n) I[ỹ]c i (i = 1,,, n) (.4) (.40)-(.41) 11
19 1. I[y] = b a F (x, y, y, y )dx y(a) = y 1, y(b) = y, y (a) = y 3, y (b) = y 4, y i (i = 1,, 3, 4) E-L F y d ( ) ( ) F + d F = 0. dx y dx y. (1) I[y] = b a y(1 + y )dx; () I[y] = b y a x dx; 3 (3) I[y] = b a (x y + y )dx. 3. E-L I[y, z] b a F (x, y(x), z(x), y (x), z (x))dx y(a) = y 1, y(b) = y, z(a) = z 1, z(b) = z, y 1, y, z 1, z 4. J(v) = 1 [ ( u ) + x u V ( ) u + y ( ) ] { } u 1 dxdydz + z Γ σu gu ds, J(u) = min v V J(v), V = C () C 1 (). 1
20 3. Green Green Laplace Dirichlet 3.1 Green R 3 Γ( )P (x, y, z), Q(x, y, z) R(x, y, z) Γ ( P x + Q y + R ) dv = (P α 1 + Qα + Rα 3 ) ds, (3.1) z Γ dv ds Γ α = (α1, α, α 3 ) = (cos( n, x), cos( n, y), cos( n, z)), (3.) nγ u, v C () C 1 (). (3.1) P = u v x, Q = u v y, R = u v z, u vdv = = x + y + z, Γ u v ( u n ds v x x + u v y y + u v z z n = α 1 x + α y + α 3 z ) dv, (3.3) Laplace n (3.3) Green (3.3) u v v udv = Γ v u ( u n ds v x x + u v y y + u v z z ) dv. (3.4) 13
21 (3.3)(3.4) (u v v u)dv = Γ ( u v n v u ) ds. (3.5) n (3.5)Green Green 3. Green v = r 1 P 0 P = [ (x x 0 ) + (y y 0 ) + (z z 0 ) ] 1/, (3.6) P 0 (x 0, y 0, z 0 ) P (x, y, z) v = 1 P 0 Laplace r P0 P u = u xx + u yy + u zz = 0. (3.7) Laplace(3.7) Laplace v P 0 v Green GreenP 0 εb ε ( \ B ε )( \ B ε ) v \ B ε uvgreen(3.5) ( u r 1 r 1 u ) ) dv = (u r 1 u r 1 ds, (3.8) \B ε n n Γ B ε B ε B ε \ B ε B ε r 1 n Bε u r 1 u = 0, r 1 = 0. (3.9) = r ( 1 r n ds = 1 ε ) = 1 r = 1 ε, B ε uds = 4πu, (3.10) uu B ε 1 u B ε r n ds = 1 u ds = 4πε u ε B ε n n, (3.11) 14
22 u n u n B ε(3.8) ε 0 Γ (u r 1 n u(p 0 ) = 1 ( ( u(p ) 4π Γ n ) u r 1 ds + 4πu 4πε u n n = 0. 1 r P0 P ) 1 r P0 P ) u(p ) ds P. (3.1) n up 0 (3.1)Γ (3.1)u P 0 P 0 Γ Γ [ u n ( ) 1 1 r r ] u ds = n 0 P 0 R 3 \, πu(p 0 ) P 0 Γ, 4πu(P 0 ) P 0, u Poisson (3.1) u(p 0 ) = 1 ( ( u(p ) 4π Γ n 1 r P0 P (3.15) ) (3.13) u = f, (3.14) 1 r P0 P ) u(p ) ds P 1 n 4π f(p ) dv P. (3.15) r P0 P (3.1)(3.15) Poisson(3.14) 3.3 v(p 0 ) 1 4π Green 15 f(p ) dv P (3.16) r P0 P
23 3.1 uγ Laplace( u ) Γ u ds = 0. (3.17) n (3.17) Γ (3.5) u v Laplace (Neumann) u = 0 (x ), u n = f (x Γ) Γ f Γ fds = ( ) u R 3 P 0 P 0 a B a u(p 0 ) = 1 uds. (3.18) 4πa B a (3.1)P 0 B a u(p 0 ) = 1 ( ( ) u(p ) 1 1 4π B a n r P0 r P P0 P ) u(p ) ds P. (3.19) n B a r P0 P = a. 3.1 B a 1 r P0 P u(p ) n ds P = 1 u(p ) a B a n ds P = 0. (3.0) ( ) 1 = ( ) 1 = 1 n r B a r r B a r = 1 Ba a, 16
24 B a u n ( 1 r (3.0)(3.1) (3.19) (3.18) ) ds = 1a u(p 0 ) = 1 uds. 4πa B a B a uds. (3.1) (3.1)(3.1) u u n u(p ) Γ B a (3.18)(3.18) a < a 0 B a (3.1)a a 0 B a0 (3.18) a 0 P 0 a B a B a0 Γ 3.1: B a B a
25 3.3 () R 3 u = u(x, y, z) u u u u m( 3.3 )up 0 m P 0 r > 0 B r rb r B r B r u m u B r P m up B r u < mu B r (3.18) 1 uds < 1 mds = m. (3.) 4πr B r 4πr B r 1 uds = u(p 4πr 0 ) = m. B r (3.) B r u mp 0 r r B r u mb r u m u = m P P 0 P 1 lb ri (i = 1,, m) m i=1b ri l B r1 P 0 B r B r1 B rn B rn 1 (3.) 18
26 P 0 B r1 P 1 B r P m P B rm 3.: l B ri B ri (i = 1,, m)u = mp u(p ) = m P u m u u 3. uu 3.3 u v u v u(x, y, z) v(x, y, z), (x, y, z). u v 3.5 Laplace Laplace(Poisson) Dirichlet( ) Dirichlet u = u x + u y + u z = 0, u = g(x), (3.3) R 3 g 19
27 3.4 g (3.3) (3.3) u 1 u (u 1 u ) 0 u = u 1 u Laplace u u 0. u 1 u. (3.3) (3.3) g 1 g g 1 g ε, (3.4) ε u 1 u g 1 g u 1 u g 1 g. max{u 1 u } = max {g 1 g } ε, min{u 1 u } = min {g 1 g } ε. max{ u 1 u } max { g 1 g } ε. (3.5) (3.3) Dirichlet u = 0, x, u = g(x), x, u 0, r = x + y + z. 3.5 g (3.6) (3.6) u 1 u (3.6)u = u 1 u u u = 0 (3.7) u(x, y, z) 0, r = x + y + z (3.8) 0
28 u P u(p ) 0 u(p ) > 0. RR B R P B R R (3.3) R c O P B R R 3.3: R c (3.8) R B R u BR < u(p ). (3.9) (3.7)(3.9) R B R u u (3.6) 1. (3.13) P 0 Γ. u(x, y) u = sin θ θu 3. Laplace Neumann fds = 0 4. Laplace Dirichlet 1
29 5. n i,j=1 a ij u + x i x j n i=1 b i u x i + cu = 0, a ij, b i, c (i, j = 1,, n) a ij λ i (i = 1,, n) n a ij λ i λ j a n i,j=1 i=1 λ i (a ), c < 0 u Γu 6. u x + u + cu = 0 (c > 0) y
30 4. Green Green Green Laplace DirichletGreen Green GreenLaplaceDirichlet Poisson Poisson 4.1 Green u u (3.1) u(p 0 ) = 1 4π Γ ( u(p ) n ( 1 r P0 P ) 1 r P0 P ) u(p ) ds P, (4.1) n P 0 (x 0, y 0, z 0 ) Γu u n Γ u(4.1) Laplace(4.1)u u n Γ (u u n ) (4.1)Laplace(3.7) Dirichlet NeumannLaplace(3.7) Dirichlet u = u xx + u yy + u zz = 0, (4.) u Γ = f, (4.3) u Γ u Γ ( 3.4 n (4.)-(4.3) u n Γ )(4.1) u n Green u n g(p, P 0 ) : P Γ(4πr P0 P ) 1 g(p, P 0 ) Γ = 1 1 4πr P0 P. (4.4) Γ
31 Green(3.5) (4.1) Γ u(p 0 ) = ( g u n u g ) ds = 0. n Γ ( G u ) n u G ds, (4.5) n G(P, P 0 ) = 1 4πr P0 P g(p, P 0 ). (4.6) G Laplace(4.) Dirichlet Green (Dirichlet ) (4.4) G = G(P, P 0 ) Γ Green G(P, P 0 )(4.5) (4.)-(4.3) ( u(p 0 ) = f G ) (P )ds P. (4.7) Γ n GreenG = G(P, P 0 ) (4.)-(4.3) (4.7)Dirichlet (4.7) (4.)(4.3) Green Green Green Laplace (4.)Dirichlet g = 0, g Γ = 1 4πr P0 P. Γ (4.8) Dirichlet Dirichlet Laplace Dirichlet Green (i) Green
32 Green Dirichlet (4.7) (ii) Green Dirichlet(iii) (4.7)Dirichlet GreenGreen 4.1 GrennG(P, P 0 ) P = P 0 Laplace(4.) P P 0 G(P, P 0 ) 1 4πr P0 P 4. Γ GreenG(P, P 0 ) < G(P, P 0 ) < 1 4πr P0 P. GrennG(P, P 0 )P P 0 P 1 P G(P 1, P ) = G(P, P 1 ). G(P, P 0 ) 4.5 ds P = 1. Γ n 4.1 Green P 0 1. P 0 4πr P0 P Green G(P, P 0 ) = 1 4πr P0 P g(p, P 0 ) g(p, P 0 ) Green 4.4 P 1 P P P 1 4. Green Green g(p, P 0 ) g Green 3
33 P 0 P 1 P 0 P 1 P 0 P 1 P 1 P 0 P 1 P 0 Γ Green Poisson Green Poisson B R O R(4.1) P 0 (x 0, y 0, z 0 ) OP 0 OP 1 r 0 r 1 = R, (4.9) r 0 = OP 0, r 1 = OP 1. P 1 P 0 B R Q P 1 R r 1 O r 0 P 0 B R 4.1 P 0 P 1 Q B R OQP 0 OQP 1 O OP 0 OQ = OQ OP 1 ( r 0 r 1 = R, (4.9) ), 4
34 OQP 0 OQP 1 Q QP 1 = R r 0 P 0 Q. (4.10) P 1 (4.10) P 0 P 1 R r 0 r P1 P g(p, P 0 ) = 1 4π R 1, r 0 r P1 P = P 1P B R Green ( ) G(P, P 0 ) = 1 1 R 1. (4.11) 4π r P0 r P 0 r P1 P Green(4.11)Laplace(4.) B R Dirichlet (4.7) G n B R r P0 P u BR = f (4.1) = r 0 + r r 0 r cos η, r P1 P = r 1 + r r 1 r cos η, r = OP η = (OP 0, OP )OP 0 OP (4.9) (4.11) Green ( G(P, P 0 ) = 1 4π 1 r 0 + r r 0 r cos η ) R. r 0 r R rr 0 cos η + R 4 B R G n = G r=r r r=r = 1 { } r r 0 cos η 4π (r + r0 r 0 r cos η) (r0r R r 0 cos η)r 3/ (r0r R rr 0 cos η + R 4 ) r=r 3/ = 1 R r0 4πR (R + r0 Rr 0 cos η). 3/ (4.7)B R Dirichlet(4.), (4.1) u(p 0 ) = 1 R r0 4πR B R (R + r0 Rr 0 cos η) f(p )ds 3/ P. (4.13) 5
35 u(r 0, θ 0, ϕ 0 ) = R 4π π π 0 0 R r0 f(r, θ, ϕ) sin θdθdϕ, (4.14) (R + r0 Rr 0 cos η) 3/ (r 0, θ 0, ϕ 0 ) P 0 (R, θ, ϕ) B R P OP 0 OP (sin θ 0 cos ϕ 0, sin θ 0 sin ϕ 0, cos θ 0 ), (sin θ cos ϕ, sin θ sin ϕ, cos θ), cos η = cos θ cos θ 0 + sin θ sin θ 0 (cos ϕ cos ϕ 0 + sin ϕ sin ϕ 0 ) = cos θ cos θ 0 + sin θ sin θ 0 cos(ϕ ϕ 0 ). (4.13) (4.14)Poisson Green Poisson Dirichletz > 0 u = u(x, y, z)z = 0f(x, y) u z=0 = f(x, y). P 0 : (x 0, y 0, z 0 ) z = 0P 0 : (x 0, y 0, z 0 )(z 0 > 0) Green [ ] G(P, P 0 ) = 1 1 4π (x x0 ) + (y y 0 ) + (z z 0 ) 1. (x x0 ) + (y y 0 ) + (z + z 0 ) (4.15) z > 0 z = 0 z- n = z. z > 0u 1 u(p ) = O( OP ), u n (P ) = O( 1 OP ) ( OP ), (3.1) () Green (4.7) (4.7) z > 0 6
36 Laplace(4.) Dirichlet u(x 0, y 0, z 0 ) = 1 4π = z 0 π [ f(x, y) z (A A + )] dxdy z=0 f(x, y) dxdy, [(x x 0 ) + (y y 0 ) + z0] 3/ (4.16) A ± = 1 (x x0 ) + (y y 0 ) + (z ± z 0 ). (4.16) z > 0 Poisson Green Poisson Laplace Dirichlet u = u xx + u yy = 0, (x, y), (4.17) u = f(θ), (4.18) = {(x, y) x + y < R }(R > 0 ) = {(x, y) x + y = R } θ GreenGreen 1 4πr 1 π ln r 1 ( 1) Green [ ( )] G(P, P 0 ) = 1 ln 1 R ln. π r P0 r P 0 r P1 P r P0 P = r 0 + r r 0 r cos η, r P1 P = r 1 + r r 1 r cos η, η = (OP 0, OP )OP 0 OP 4.. OP 0 OP (cos θ 0, sin θ 0 ) (cos θ, sin θ) cos η = cos(θ θ 0 ) = cos θ cos θ 0 + sin θ sin θ 0. r 1 r 0 = R 7
37 r = R G n = G r=r r { r=r [ ( 1 = ln r π = 1 { π = 1 πr 1 r 0 + r r 0 r cos η ln r r 0 cos η r + r 0 rr 0 cos η R r0. R Rr 0 cos η + r0 R r 0 r R rr 0 cos η + R 4 )]} r=r r 0r R r 0 cos η r 0r R rr 0 cos η + R 4 } r=r P P 1 R O η P r 0 0 r 1 4. P 0 P 1 (4.7) Dirichlet(4.17)-(4.18) u(r 0, θ 0 ) = 1 R r0 f(θ)ds πr x +y =R R Rr 0 cos η + r0 = 1 π (R r0)f(θ) dθ. π 0 R Rr 0 cos(θ θ 0 ) + r0 Poisson (4.19) 4.3 Laplace Dirichlet Poisson (4.19)u(4.17). 8
38 GreenG(P, P 0 )P, P 0 P G(P, P 0 ) P 0 (4.19) u(4.17) P 0 P (4.19) G n u(4.18) (r 0, θ 0 ) (R, θ) (4.19) u(r 0, Q 0 ) f(θ). (4.19)ϕ = θ θ 0 f(θ)cos(θ θ 0 ) θ π(4.19) u(r 0, θ 0 ) = 1 π Green ( 4.5 ) 1 π u(r 0, θ 0 ) f(θ) = 1 π π π π π π π (R r0)f(ϕ + θ 0 ) dϕ. (4.0) R Rr 0 cos ϕ + r0 R r0 dϕ = 1. R Rr 0 cos ϕ + r0 R r0 [f(ϕ + θ R Rr 0 cos ϕ + r0 0 ) f(θ)]dϕ. (4.1) r 0 R θ 0 θ f(θ)ε δ ϕ [ δ, δ] θ θ 0 [ π, π] f(ϕ + θ 0 ) f(θ) ε. ( π, δ), ( δ, δ), (δ, π), (4.1)I 1 I I 3 u(r 0, θ 0 ) f(θ) = I 1 + I + I 3, u(r 0, θ 0 ) f(θ) I 1 + I + I 3. I I 1 I 3 9
39 ( π, δ) cos ϕ cos δ R Rr 0 cos ϕ + r 0 R Rr 0 cos δ + r 0 = (R r 0 ) + Rr 0 (1 cos δ) 4Rr 0 sin δ. f f(ϕ + θ 0 ) f(θ) K 1, K 1 I 1 = 1 π δ π R r0 [f(ϕ + θ R Rr 0 cos ϕ + r0 0 ) f(θ)]dϕ K 1(R r0)(π δ) 8πRr 0 sin δ. (4.) I 3 I 3 K 3(R r0)(π δ) 8πRr 0 sin δ, (4.3) K 3 (4.) (4.3) r 0 RI 1 I 3 II I I = 1 π δ ε 1 π ε 1 π δ δ R r 0 R Rr 0 cos ϕ + r 0 0, R r0 [f(ϕ + θ R Rr 0 cos ϕ + r0 0 ) f(θ)]dϕ δ π π R r0 dϕ R Rr 0 cos ϕ + r0 R r0 dϕ = ε R Rr 0 cos ϕ + r0. ε(r 0, θ 0 ) (R, θ)u(r 0, θ 0 )f(θ) (4.4) u(r 0, θ 0 ) f(θ). 10
40 (4.19) u Dirichlet (4.17)- (4.18) 1. Green Green G(P 1, P ) = G(P, P 1 ). 3. Poisson u xx + u yy + u zz = 0 (x + y + z < 1), u(r, θ, ϕ) r=1 = 3 cos θ + 1 (r, θ, ϕ ) 4. u a C (1) u C = A cos ϕ; () u C = A + B sin ϕ. 5. LaplaceDirichlet u = u xx + u yy = 0 (y > 0), u y=0 = f(x) 11
41 5. ( ) Poisson (4.13) Poisson 5.1 u(p ) = u(x, y, z)q QQ lim {r u(p )} = 0, (5.1) P Q P Q r P Q = P Q P Q u Q u(p )Q ( Q) Q B R QRQ B R LaplaceB R Laplace B R B R u Poissonũ B R Q u ũ. uq ũq v = u ũ. v(i)b R Q (ii) Q(5.1) (iii) B R v B R \{Q} lim {r v(p )} = 0; (5.) P Q P Q ( 1 v ε (P ) = ε r QP v 0. (5.3) 1 R ), (5.4) ε > 0v ε (a) B R v ε 0 (b) B δ B R (5.1) δ B δ Q δ B δ. 1
42 B δ δ Q R B R 5.1 B R B δ ε (5.)δ > 0 B δ v v ε, B R v = v ε 0. ( 3.3)P v(p ) v ε (P ). (5.5) P ε 0 v(p ) = 0.P v 0.δ > 0B R \{Q} (5.3) Poisson 5. u(p 0 )x 0, y 0, z 0 P0 : (x 0, y0, z0)ux 0 x 0, y 0 y0, z 0 z0 P0 P0 BB R. u(p 0 ) ub Poisson u 13
43 B u(p 0 ) = 1 4πR = 1 4πR B B R r0 u(p ) (R + r0 Rr 0 cos η) ds 3/ P u(p ) R r0 ds r 3 P. P 0 P P 0 u(p 0 ) P 0 B (R r 0)r 3 P 0 P = [R (x 0 + y 0 + z 0)][(x x 0 ) + (y y 0 ) + (z z 0 ) ] 3/ (5.6) : (0, 0, 0)x 0, y 0, z 0 = [R (x 0 + y0 + z0)][x + y + z (xx 0 + yy 0 + zz 0 ) + x 0 + y0 + z0] 3/ [ = [R (x 0 + y0 + z0)] {R 1 (xx ]} 0 + yy 0 + zz 0 ) (x 0 + y0 + z0) 3/, R (5.7) x 0, y 0, z 0 (x, y, z) B (x 0, y 0, z 0 ) x 0, y 0, z 0 u(p 0 ) P 0 = P 0 P 0 u(p 0 ) Poisson 5.3 {u k } {u k }u f k = u k. (5.8) f k {f k } ε > 0 N N n, m > N f n f m ε. n, m u n u m ε. Cauchy {u k }u. {u k } uu 14
44 P 0 P 0 B. B u k Poisson(4.14) π π u k (r 0, θ 0, ϕ 0 ) = R u k (R, θ, ϕ) 4π 0 0 (R r0) sin θ dθdϕ, {R + r0 Rr 0 [cos θ cos θ 0 + sin θ sin θ 0 cos(ϕ ϕ 0 )]} 3/ (5.9) RB(r 0, θ 0, ϕ 0 )P 0 {u k (R, θ, ϕ)} u(5.9) k π π u(r 0, θ 0, ϕ 0 ) = R u(r, θ, ϕ) 4π 0 0 (R r0) sin θ dθdϕ. {R + r0 Rr 0 [cos θ cos θ 0 + sin θ sin θ 0 cos(ϕ ϕ 0 )]} 3/ (5.10) upoisson ub Harnack {u k }P u P R B R. Q B R B R upoisson(4.13) u(q) = 1 R r0 4πR B R (R + r0 Rr 0 cos η) u(m)ds M. 3/ u 0u P u(q) R r 0 (R + r 0 ) 3 R r 0 (R + r 0 Rr 0 cos η) 3/ R r 0 (R r 0 ) 3, 1 u(m)ds M R r 0 1 u(q) u(m)ds 4πR B R (R + r 0 ) M R + r 0 4πR B R (R r 0 ). R r 0 (R + r 0 ) R u(p ) u(q) R + r 0 R u(p ). (5.11) (R r 0 ) 15
45 {u k } v k (5.11) v k = u k u k 1 0 (k =, 3, ). R r 0 (R + r 0 ) R v k(p ) v k (Q) R + r 0 (R r 0 ) R v k(p ). (5.1) m n(n > m) u n u m = n k=m+1 v k. (5.1) u k (Q) B R 5.3 {u k (Q)} B R u(q). {u k (P )} M lp Ml {u k } M {u k } u. D B i (i = 1,,, m)dd m i=1b i. {u k } B i P i {u k } B i (i = 1,,, m) D 5.4Harnack 5.5 (5.11) Harnack ud DC max{u} C min{u}. (5.13) D D D r. Dm r 0 r 0 r (r r 0 )r (r + r 0 ) 9, (r + r 0 )r 6. (5.14) (r r 0 ) (5.11) D r P 1 P 9 u(p 1) 6u(P ), u(p 1 ) 7u(P ). (5.15) 16
46 u D r C = (7)m (5.13) 1. Q u(p ) Q ( u(p ) = o ln 1 ), r P Q u(p )Q Q. u(p ) Q h α (0, 1] r α QP hp Qu 1 r QP α = 1. 17
47 6. Laplace u 0 n n ( ) B R R u = u(x, y, z)b R B R (x, y, z) u(x, y, z) > u(x 0, y 0, z 0 ), (6.1) P 0 : (x 0, y 0, z 0 ) B R u(x, y, z)p 0 γ γ P 0 P 0 u γ > 0. (6.) B R up 0 P 0 u 0. γ a v(x, y, z) = exp{ a(x + y + z )} exp{ ar }, (6.3) (i) B R x + y + z = R v 0 (ii) { } (x, y, z) R 4 x + y + z R v a v > 0. v = [ 6a + 4a (x + y + z ) ] exp{ a(x + y + z )}. 1
48 x + y + z 1 4 R, a (a > 6/R ) v > 0. v (iii) v dv r > 0 dv dr dr < 0 B R : x + y + z = R P 0 v γ = dv cos ( γ, r) > 0. (6.4) dr w(x, y, z) = u(x, y, z) εv(x, y, z) u(x 0, y 0, z 0 ). (6.5) ε > 0 w(x, y, z) 0, (x, y, z). (6.6) x + y + z = R 4 u(x, y, z) > u(x 0, y 0, z 0 ) ε > 0 w w > 0 x + y + z = R w > 0. w = ε v < 0, w = B R B R w 0 (6.6) (6.5)(6.6) w(x, y, z) 0 = w(x 0, y 0, z 0 ), (x, y, z). P 0 (6.4) w γ 0, u γ ε v γ u γ 0. ε v γ > 0.
49 (6.) 6. Γ P B P P Γ u(x, y, z)p 0 Γ () P 0 u u n n ( ) P0 B P0 P 0 Γ u(x, y, z) u B P0 u P u n P 0 Γ u u ( u) n < 0 u > 0. n 6. Laplace Laplace Laplace Neumann u = u x + u y + u z = 0 u n = g. (6.7) (6.7) u (6.7)u + c(c )(6.7) (6.7) Laplace Neumann u 1 u (6.7)u = u 1 u u = u n = 0. u 3
50 (P 0 )6.uP 0 u n < 0. P0 u n = 0 u Laplace Neumann R 3 P ( c = R 3 \ ) B P 6.1 c B P P c 6.1. c. u = u x + u y + u z = 0, x R3 \, u ν = g, u 0 ( r = (6.8) x + y + z ), ν c g 6.4 (6.8) (6.8)u 1 u. u = u 1 u u ν = 0, (6.9) u c c r u u 0. uε > 0 RR B R P u(p ) ε. (6.10) 4
51 B R R. R u. u R 6.u u B R (6.10) R u ε. ε c u Laplace C () C 1 () Poisson u C () C 1 () u = 0 u n = 0 u E(u) = 1 [ ( u ) ( ) ( ) ] u u + + dxdydz x y z = 1 [ x ( u u x ) + y ( u u y u Green E(u) = 1 ) + z u u n ds. ( u u ) ] u u dxdydz. z u = 0 u n = 0. E(u) = 0, u x = u y = u z = 0, u. (6.11) u = 0 u 0, Poisson C () C 1 () (6.11)Poisson 5
52 ( σu) u n + = f (σ > 0) 3. (i) (iii) v(x, y, z) x + y + z R v > 0. 6
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1 2 (beta 10 ) 3 SOLVED AND TEXIFIED BY 4 HONORED REVIEWER BBS (lilybbs.us) 1 2002 6 1 2003 1 2 2 ( ) (E-mail: xiaoxinpan@163.com) 3 beta 2005 11 9 ( / ) 40.97% 4 02CS chouxiaoya tedy akaru yitianxing