拉普拉斯(Laplace)方程

LaplaceLaplace()Poisson()LaplacePoissonLaplacePoisson()GreenLaplaceGreen()LaplacePoissonHarnackLaplaceLaplacex1::::::::::::Laplace:::(::::::::::)4u,nXi=1@2u@x2i=0(1.1)Poisson4u,nXi=1@2u@x2i=f(x1;¢¢¢;xn)(1.2)u=u(x1;¢¢¢;xn)f(x1;¢¢¢;xn)n=2;31.1LaplacePoisson1Gaussdiv¡!E=10½(x;y;z);(1.3)¡!E(x;y;z)0½(x;y;z)r£¡!E=0(curl¡!E=0);(1.4)¡!E=¡ru(x;y;z);(1.5)u::::::(::::::::)(1.5)(1.3)4u(x;y;z)=¡10½(x;y;z):(1.6)(1.6):::::::::::Poisson:::­½´0:::::::::::Laplace:::4u=0:(1.7)LaplaceP0:(x0;y0;z0)mP:(x;y;z)¡!F(x;y;z)mr2¡¡!PP0P0r=p(x¡x0)2+(y¡y0)2+(z¡z0)2P0P¡!F(x;y;z)¡!F(x;y;z)=¡mr2µx¡x0r;y¡y0r;z¡z0r¶:(1.8)¡!F:::::::::::::(1.8)¡!Fu(x;y;z)=mr(1.9)¡!F=ru:2­½(x;y;z)­­u(x;y;z)=ZZZ­½(»;´;³)d»d´d³p(x¡»)2+(y¡´)2+(z¡³)2:(1.10)u­Laplace4u=uxx+uyy+uzz=0;(1.11)½(x;y;z)HÄolderu­Poisson4u=¡4¼½:(1.12)(1.11)(1.12)u4u=8:0;(x;y;z)2Rn­;¡4¼½;(x;y;z)2­½(x;y;z)HÄolder(1.13)½@2u@t2=Tµ@2u@x2+@2u@y2¶+F(t;x;y):(1.14)F(x;y)ut(1.14)@2u@x2+@2u@y2=¡F(x;y)T:(1.15)(1.15)PoissonPoissonLaplaceLaplaceLaplace(1.1)Poisson(1.2)31.1Laplace(1.1)(1.1):::::::::::¤1.1¤1.2­(1.1)(1.2)()t::::::::::::::::::::(1.1)(1.2)(Dirichlet)Rn­@­gu=u(x1;¢¢¢;xn)­(1.1)((1.2))­@­uj@­=g:¤(1.16)(1.16):::::::::::::::::::::::::::::::Dirichlet::::::::(Neumann)¡gu=u(x1;¢¢¢;xn)¡­(1.1)((1.2))­¡¡n@u@n@u@n¯¯¯¯¡=g:¤(1.17)(1.17)::::::::::::::::::::::::::::::::Neumann::::::::DirichletNeumann­()uLaplace(1.1)(n=3)uj@­=gg­4~u=r'~u''­Laplace(1.1)(n=3)@­@'@n¯¯¯¯@­=0:­@­Laplace@­:::::::::::Laplace:::::::::::::::::::::::::Dirichlet::::::::::::::::::Neumann:::::LaplaceLaplace¡¡Dirichletuj¡=1:u1´1u2=(x2+y2+z2)¡1=2Laplacelimr!1u(x;y;z)=0(r=px2+y2+z2):(1.18)(1.18)u1´1:LaplaceDirichletR3¡gu(x;y;z)(i)¡QLaplace(ii)Q[¡(iii)(1.18)(iv)¡uj¡=g:¤(1.19)Neumann¡gu(x;y;z)(i)¡QLaplace(ii)Q[¡(iii)(1.18)(iv)¡Q~n(¡)@u@~n@u@~n¯¯¯¯¡=g:¤(1.20)51.2DirichletNeumann¤1.3¤1.4nLaplace¤1.5Poisson(1.2)PoissonLaplace(1.1)fHÄolder(n=3(1.10))Laplace(1.1)¤1.u(x1;¢¢¢;xn)=f(r)(r=px21+¢¢¢+x2n)n(ux1x1+¢¢¢+uxnxn=0)f(r)=c1+c2rn¡2(n6=2);f(r)=c1+c2ln1r(n=2);c1;c22.(r;µ;')4u=1r2@@rµr2@u@r¶+1r2sinµ@@µµsinµ@u@µ¶+1r2sin2µ@2u@'2:3.(r;µ;z)4u=1r@@rµr@u@r¶+1r2@2u@µ2+@2u@z2:4.(1)ax+by+c(a;b;c);(2)x2¡y22xy;(3)x3¡3xy23x2y¡y3;(4)shnysinnxshnycosnxchnysinnxchnycosnx(n);6(5)shxchx+cosysinychx+cosy:5.Laplace(1)lnrµ;(2)rncosnµrnsinnµ(n)(3)rlnrcosµ¡rµsinµrlnrsinµ+rµcosµ:6.Laplace(06x6a;06y6b)8:uxx+uyy=0;u(0;y)=u(a;y)=0;u(x;0)=sin¼xa;u(x;b)=0:7.LaplaceDirichletu(x;y)7x2:()::::::::::LaplacePoissonLaplacePoissonLaplace(Poisson)¤()::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::(::::::::::::::::::::::)::::::::::2.1Euler-LagrangeHamiltonqi(t)(i=1;¢¢¢;n)_qi(t)(i=1;¢¢¢;n)LagrangeL(t;qi;_qi)[t1;t2]t():::::±Zt2t1L(t;qi;_qi)dt=0:(2.1)±qijt=t1=0;±qijt=t2=0(i=1;¢¢¢;n)(2.1):::::::::::::Lagrange:::::::::::@L@qi¡ddtµ@L@_qi¶=0(i=1;¢¢¢;n):(2.2)(2.2)(2.1)y=f(x)xxf()f0(x)=0(¤1f00(x)()f(x)())y=y(x)[a;b]()I[y],ZbaF(x;y(x);y0(x))dx;(2.3)F(2.3)Fy=y(x)I[y]y(x)::::::y(x)I[y]y(x)F(x;y;y0)y(x)xy(a)=y¤;y(b)=y¤(2.4)y¤y¤(2.4)I[y]y(x)y(x)±y(x)y(x)+±y(x)±y(x)y(x)F(x;y+±y;y0+±y0)=F(x;y;y0)+@F@y±y+@F@y0±y0;(2.3)I[y+±y]¡I[y]=Zbaµ@F@y±y+@F@y0±y0¶(x)dx:I[y]±I[y]±I[y]=Zbaµ@F@y±y+@F@y0±y0¶(x)dx:(2.5)2.1I[y]@F@y¡ddxµ@F@y0¶=0:¤(2.6)2y=y(x)I[y]y(x)±y(x)=´(x);(2.7)´(x)´(a)=´(b)=0:(2.8)(2.8)y+´(2.4)2.1.-6y(x)y(x)+´(x)abx0y¤y¤y2.1:y(x)y(x)+´(x)I[y+´(x)]=ZbaF(x;y(x)+´(x);y0(x)+´0(x))dx(2.9)=0y(x)+´(x)=y(x)y(x)I[y]=0I[y+´(x)]dI[y+´]d¯¯¯¯=0=0:(2.9)Zbaµ@F@y´+@F@y0´0¶(x)dx=0:(2.10)(2.8)(2.10)0=Zbaµ@F@y´¶(x)dx+@F@y0´¯¯¯¯x=bx=a¡Zba·ddxµ@F@y0¶´¸(x)dx=Zba·@F@y¡ddxµ@F@y0¶¸(x)´(x)dx:3´(x)@F@y¡ddxµ@F@y0¶=0:2.1.¥(2.6)::::::::::::::::::::Euler-Lagrange:::::::::E-L:::nu(x1;¢¢¢;xn)I[u]=Z¢¢¢Z­F(x1;¢¢¢;xn;u(x1;¢¢¢;xn);ux1(x1;¢¢¢;xn);¢¢¢;uxn(x1;¢¢¢;xn))dx1¢¢¢dxn;(2.11)­Rn­@­uuj@­=g(x1;¢¢¢;xn);8(x1;¢¢¢;xn)2@­:(2.12)2.12.2I[u]@F@u¡nXi=1@@xiµ@F@uxi¶=0:¤(2.13)u=u(x1;¢¢¢;xn)I[u]u(x),u(x1;¢¢¢;xn)±u(x)u(x)+±u(x)±u(x)u(x)±u(x)=´(x)´(x)=´(x1;¢¢¢;xn)´j@­=0:(2.14)(2.14)u+´(2.12)dI[u+´]d¯¯¯¯=0=0:(2.11)Z¢¢¢Z­@F@u´+nXi=1@F@uxi´xi#(x1;¢¢¢;xn)dx1¢¢¢dxn=0:(2.15)4Green(2.14)(2.15)0=Z¢¢¢Z­@F@u´dx1¢¢¢dxn+Z@­´(rDuF¢~n)ds¡Z¢¢¢Z­´nXi=1@@xi¡Fuxi¢dx1¢¢¢dxn=Z¢¢¢Z­´@F@u¡nXi=1@@xi¡Fuxi¢#dx1¢¢¢dxn;(2.16)´=´(x)(2.14)­rDuF=(Fux1;¢¢¢;Fuxn)~n@­´(2.16)@F@u¡nXi=1@@xiµ@F@uxi¶=0:(2.17)(2.13)¥(2.13)(2.11):::::::::::::::::::Euler-Lagrange::::::::::E-L::FF=12nXi=1µ@u@xi¶2;E-LLaplace4u=nXi=1@2u@x2i=0:2.1I[u1;¢¢¢;um]=Z¢¢¢Z­F(x;u1(x);¢¢¢;um(x);Du1(x);¢¢¢;Dum(x))dx1¢¢¢dxn;x=(x1;¢¢¢;xn)Dui=µ@ui@x1;¢¢¢;@ui@xn¶::::::::::::::::::::Euler-Lagrange:::::uu¤(2.3)y(a)=y0;y(b);(2.18)5y0x=by(2.3)y=y(x)x=bI[y]x=by(b)y=y(x)I[y](2.3)F(x;y(x);y0(x))E-L(2.16)±y(x)=´(x)´(x)´(a)=0:(2.19)(2.16)dI[y+´]d¯¯¯¯=0=0@F@y0´(x)¯¯¯¯ba+Zba·@F@y¡ddxµ@F@y0¶¸´(x)dx=0:(2.20)(2.19)E-L(2.16)´(b)(2.20)@F@y0¯¯¯¯x=b=0:(2.21)(2.21)::::::::::::::::by(b)y0(b)I[u]=ZZ­F(x;y;u(x;y);ux(x;y);uy(x;y))dxdy+Z@­G(s;u;us)ds;(2.22)­R2s@­us=@u@sFG2.22.3I[u]((2.22))@F@u¡@@xµ@F@ux¶¡@@yµ@F@uy¶=0(2.23)@­·@F@uxdyds¡@F@uydxds+@G@u¡ddsµ@G@us¶¸¯¯¯¯@­=0:¤(2.24)62.32.2±u(x;y)@­(2.24)(2.22)FGF=u2x+u2y;G=¾u2;(2.25)¾@­uE-L4u=uxx+uyy=0;(2.26)µuxdyds¡uydxds+¾u¶¯¯¯¯@­=0;µ@u@n+¾u¶¯¯¯¯@­=0;(2.27)n@­