偏微分方程简明教程

2008.9-2008.12²Laplace²Li-Yau'sHarnackinequality²CauchyCauchy-Kowalevski|H.Lewy²²²Openproblems²F.John,PartialDi®erentialEquations,Springer-Verlag,1982.²2002²20081x1:²PDEu(x;y;¢¢¢)PDEF(x;y;¢¢¢;u;ux;uy;¢¢¢;uxx;uxy;¢¢¢)=0(1)Fx;y;¢¢¢;uu:::::²(solution)u(1):::u(x;y;¢¢¢)(1)x;y;¢¢¢­(1)|x;y;¢¢¢u(1)u­x;y;¢¢¢­x;y;¢¢¢²(PDEs):nmPDEnm(under-determined)nm(over-determined)²PDEPDEs²PDEPDEsx;y;¢¢¢²::::::::PDE::::::::::ux;y;¢¢¢:::::::::::::mPDE:::::::::::::::ummx;y;¢¢¢um2:::::::::::::mPDE::::::::::::::::::um|PDE8:(linear)(nonlinear)8:(quasilinear)(fullynonlinear)|::x2:PDEst(x1;x2;¢¢¢;xn)n=3x;y;zn¢,@2@x21+¢¢¢+@2@x2n=nXi=1@2@x2iLaplace,¤,@2@t2¡@2@x21¡¢¢¢¡@2@x2n=@2@t2¡nXi=1@2@x2i=@2@t2¡¢1.Laplace¢u,@2u@x21+¢¢¢+@2u@x2n=nXi=1@2u@x2i=0:(2)u::::::::::::::::::(harmonicfunction)|n=2x1=x;x2=yv(x;y)uvCauchy-Riemannux=vy;uy=¡vx(3)(3)(u;v)z=x+iyf(z)=f(x+iy)=u(x;y)+iv(x;y):(4)3|(u(x;y);¡v(x;y))|n=3(2)2.waveequationutt=c2¢u(c0);(5)u=u(t;x1;¢¢¢;xn)|n=1:cn=2:n=3:3.MaxwellMaxwellequationsE=E(E1;E2;E3)H=(H1;H2;H3)Maxwell..8:Et=curlH;¹Ht=¡curlE;divE=divH=0;(6);¹Et=curlH;¹Ht=¡curlEt=0divE=divH=0tEi;Hkc2=1=¹(5)(6)curl(curlE)=¡¹(curlH)t=¡¹Ett;4curl(curlE)=r(divE)¡¢E;(6)Ett=(¹)¡1¢E:H4....½@2ui@t2=¹¢ui+(¸+¹)@@xi(divu)(i=1;2;3)(7)ui(t;x1;x2;x3)u½¸;¹Lameuiµ@2@t2¡¸+2¹½¢¶µ@2@t2¡¹½¢¶ui=0:(8)..ut=0.....¢2u=0:(9)5......ut=k¢u;(10)k06.V(x;y;z)mSchrÄodingeri~Ãt=¡~22m¢(Ã)+VÃ;(11)h=2¼~Planck7.Tricomiuxx=xuyy:(12)uxx=yuyy:5::::::::::::::::8.3Euclid....z=u(x;y)(1+u2y)uxx¡2uxuyuxy+(1+u2x)uyy=0:(13)9.1+nMinkowski..x=x(t;µ)2Rnjxµj2xtt¡2hxt;xµixtµ+(jxtj2¡1)xµµ=0:(14)10.½...Á(x;y)(Áx;Áy)(1¡c¡2Á2x)Áxx¡2c¡2ÁxÁyÁxy+(1¡c¡2Á2y)Áyy=0;(15)cq=pÁ2x+Á2yp=A½°(16)°c2=1¡°¡12q2:(17)11.Navier-Stokes..ukp8:@ui@t+3Xk=1@ui@xkuk+1½@p@xi=¹¢ui(i=1;2;3);3Xk=1@uk@xk=0(divu=0);(18)½¹612.8:@½@t+3Xj=1@@xj(½vj)=0;@@t(½vi)+3Xj=1@@xj(½vivj+±ijp)=0;@@t(½E)+3Xj=1@@xj(½vjE+pvj)=0;(19)½(t;x)v=(v1(t;x);v2(t;x);v3(t;x))pE=E(t;x)½pTEp=p(½;E)(p=p(½;T))(20)(20):::::::::::(20)(19)13.u(t;x)Korteweg-deVries..ut+cuux+uxxx=0;(21)14.Monge-Ap¶ereS¿¿=S2¿µ¡1Sµµ+S:(22)...::::::::::7x1:u=u(t;x)ut+cux=0(1.1)c0(t;x)-::::::::::dxdt=c:(1.2)x¡ct=const:,»;(1.3)(1.1)ududt=ddtu(t;ct+»)=ut+cux=0:(1.4)u»(1.1):::::u(t;x)=u(0;»),f(»)=f(x¡ct);(1.5)f(»)u...uu(0;x)=f(x)(1.6)fC1(R)(1.5)(1.1)fu(t;x)f1»=x¡ct»(x;t)x-u(t;x)....»»..(1.3)u(t;x)1.1-6t(t;x)x0»x¡ct=»1.1:t(x;u)-ut=Tt=0x-cTu(x;0)=u(x+cT;T)=f(x):(1.7)c.1.2--6ucxxx+cTu(0;x)u(T;x)1.2:|2......xhtk(t;x)-xhtk(t;x)v(t+k;x)¡v(t;x)k+cv(t;x+h)¡v(t;x)h=0(1.8)(1.1)h;k!0vt+cvx=0:h;k(1.8)v(0;x)=f(x)(1.9)v(1.1)(1.6)¸=k=h:(1.8)v(t+k;x)=(1+¸c)v(t;x)¡¸cv(t;x+h):(1.10)tvt+kv....EEf(x)=f(x+h):(1.11)(1.10)v(t+k;x)=((1+¸c)¡¸cE)v(t;x);(1.12)t=nk(1.8)v(t;x)=v(nk;x)=((1+¸c)¡¸cE)nv(0;x)=nXm=0Cmn¡1+¸c¢m(¡¸cE)n¡mf(x)=nXm=0Cmn¡1+¸c¢m(¡¸c)n¡mf(x+(n¡m)h):(1.13)3v(t;x)=v(nk;x)x-x;x+h;x+2h;¢¢¢;x+nh=x+t¸;(1.14)xx+nh»=x¡ct=x¡c¸nh[x;x+nh]h;k!0vv(t;x)u(t;x)f(»)u(t;x)f[x;x+t¸¡1]Courant-Friedrichs-Lewy..:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::(1.8)....f(1.13)ffv(t;x)=v(nk;x)nXm=0Cmn(1+¸c)m(¸c)n¡m=(1+2¸c)n(1.15)¸vtnv(t+k;x)¡v(t;x)k+cv(t;x)¡v(t;x¡h)h=0;(1.16)v(t+k;x)=((1¡¸c)+¸cE¡1)v(t;x):(1.17)(1.17)v(t;x)=v(nk;x)=nXm=0Cmn¡1¡¸c¢m(¸c)n¡mf(x¡(n¡m)h):(1.18)v(t;x)fx;x¡h;x¡2h;¢¢¢;x¡nh=x¡t¸(1.19)4x¡t¸¡1xh;k!0¸(1.19)x[x¡t¸;x]»=x¡ct¸¸c·1(1.20)Courant-Friedrichs-Lewy(1.20)(1.18)fv(t;x)=v(nk;x)nXm=0Cmn(1¡¸c)m(¸c)n¡m=((1¡¸c)+¸c)n=:(1.21)(1.20)fh;k!0k=h=¸(1.18)vu(t;x)=f(x¡ct)u(t;x)ju(t+k;x)¡(1¡¸c)u(t;x))¡¸cu(t;x¡h)j=jf(x¡ct¡ck)¡(1¡¸c)f(x¡ct)¡¸cf(x¡ct¡h)j·Kh2;(1.22)K=12(c2¸2+¸c)supjf00j:(1.23)fx¡ctTaylorw=u¡vjw(t+k;x)¡(1¡¸c)w(t;x))¡¸cw(t;x¡h)j·Kh2:(1.24)¸c·1,supxjw(t+k;x)j·(1¡¸c)supxjw(t;x)j+¸csupxjw(t;x¡h)j+Kh2=supxjw(x;t)j+Kh2:(1.25)w(x;0)=0(1.25)t=nkju(t;x)¡v(t;x)j·supxjw(nk;x)j·supxjw(0;x)j+nKh2=Kth¸:(1.26)5h!0w(t;x)!0(1.16)vu1.f¸·c¡1h!0(1.16)fvu.fuv2.(1.17)vjv(t+k;x)¡(1¡¸c)v(t;x)¡¸cv(t;x¡h)j±:(1.20)v(0;x)=f(x)±(1.23)Kju(t;x)¡v(t;x)j·Kth¸+t¸h±:u(t;x)¸h.3.f(x)=e®x®t;x¸=k=hn!1(1.13)(1.18)e®(x¡ct)Courant-Friedrichs-Lewyf»6x2:BurgersBurgers@u@t+u@u@x=0(2.1)Burgers(2.1)u(0;x)='(x)(2.2)Cauchy'(x)x2RC1C1Cauchy(2.1)-(2.2)x=X(t)dX(t)dt=u(t;X(t))(2.3)U(t),u(t;X(t))(2.4)X=X(t)dUdt=ut+uxdXdt=ut+uux=0,(2.5)(2.3)(2.1)(X(t);U(t))8:dXdt=U(2.6)dUdt=0(2.7)(2.6)-(2.7)(X;U)=(X(0)+tU(0);U(0)),(2.8)1X(t)=X(0)+tU(0)U(t)=U(0)(2.9)X(0)=®U(0)=u(0;X(0))='(®)(2.10)(2.9)X(t)=®+t'(®),U(t)='(®)(2.11)(t;x)x=®+t'(®)(2.12)®®=®(t;x)(2.13)(2.13)U(t)='(®)Cauchy(2.1)-(2.2)u(t;x)='(®(t;x))(2.14)(2.11)Cauchy(2.1)-(2.2)'(x)=sinx(2.15)t2[0;1)x=®+tsin®®®=®(t;x)Cauchy(2.1)-(2.2)u(t;x)=sin®(t;x)8(t;x)2[0;1)£R(2.16)'(x)=tanhxt2R+x=®+ttanh®®=~®(t;x)Cauchy(2.1)-(2.2)u(t;x)=tanh~®(t;x)8(t;x)2R+£R2::::::::::::::x=X(t):::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::'0(x)0;8x2R(2.17)t2R+(2.12)®=®(t;x)(2.17),Cauchy(2.1)-(2.2)(2.17)Cauchy(2.1)-(2.2)(2.17)t2£0;k'0(x)k¡1C0¢x®=1+'0(®)t1¡k'0(x)kC0t0(2.18)t2£0;k'0(x)k¡1C0¢(2.12)®=®(t;x)Cauchy(2.1)-(2.2)£0;k'0(x)k¡1C0¢£R(2.17)Cauchy(2.1)-(2.2)(2.17)®1®2(®1®2)'(®1)'(®2)(2.19)(0;®1)(0;®2)X1(t)=®1+t'(®1);X2(t)=®2