evans_solutions.pdf

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1、Authors:JoeBenson,DenisBashkirov,MinsuKim,HelenLi,AlexCsarEvansPDESolutions,Chapter2Joe:1,2,11;Denis:4,6,14,18;Minsu:2,3,15;Helen:5,8,13,17.Alex:10,16Problem1.Writedownanexplicitformulaforafunctionusolvingtheinitial-valueproblem(u+b
Du+cu=0onRn(0;1)tu=gonRnft=0gHe

2、rec2Randb2Rnareconstants.Sol:Fixxandt,andconsiderz(s):=u(x+bs;t+s)Thenz˙(s)=b
Du+ut=cu(x+bs;t+s)=cz(s)Therefore,z(s)=Decs,forsomeconstantD.WecansolveforDbylettings=t.Then,z(t)=u(xbt;0)=g(xbt)ct=Dei.e.D=g(xbt)ectThus,u(x+bs;t+s)=g(xbt)ec(t+s)andsowhens=0,w

3、egetu(x;t)=g(xbt)ect.Problem2.ProvethatLaplace’sequation
u=0isrotationinvariant;thatis,ifOisanorthogonalnnmatrixandwedefinev(x):=u(Ox)(x2R)then
v=0.Solution:Lety:=Ox,andwriteO=(aij).Thus,v(x)=u(Ox)=u(y)Pnwhereyj=i=1ajixi.Thisthengivesthat@vXn@u@yj=@xi@yj@xij=1Xn@

4、u=aji@yjj=112Thus,2@v3232@u36666@x177776666a11:::an177776666@y177776666:::7777=6666::::::77776666:::7777664@v7756475664@u775@xna1n:::ann@yn2@u36666@y17777=OT6666:::777766774@u5@ynTDx
v=ODy
uNow,
v=Dxv
DxvTT=(ODyu)
(ODyu)TTT=(ODyu)ODyuTTTT=(Dyu)(O)ODyuTT=(Dyu)OODyuT=

5、(Dyu)DyubecauseOisorthogonal=(Dyu)
(Dyu)=
u(y)=0Problem3.Modifytheproofofthemeanvalueformulastoshowforn3thatZZ1111u(0)=gdS+fdx;n(n)rn1@B(0;r)n(n2)(n)B(0;r)jxjn2rn2provided8>><
u=finB0(0;r)>>:u=gon@B(0;r).Solution:SetZ1(t)=u(y)dS(y);0t

6、ZZ11(r)=u(y)dS(y)=gdS:n(n)rn1@B(0;r)n(n)rn1@B(0;r)Then,ZZZt1t110(t)=
u(y)dy=fdy=fdy:n(n)tnB(0;t)n(n)tnB(0;t)(n)tn1B(0;t)(SeetheproofofThm2)3Let>0begiven.ZrZZr1(1)()=(r)0(t)dt=gdS0(t)dt:n(n)rn1@B(0;r)Usingintegrationbyparts,wecomputeZrZrZ10(t)

7、dt=fdydtn(n)tn1B(0;t)ZrZ11=fdydtn(n)tn1B(0;t)111ZrZr11Z=fdyfdSdtn(n)2ntn2B(0;t)2ntn2@B(0;t)1Zr1Z1Z1Z=fdSdtfdy+fdyn(n2)(n)tn2@B(0;t)rn2B(0;r)n2B(0;)Z11=:Ifdy+J:n(n2)(n)rn2B(0;r)ObservethatZ12J:fdyC
;forsomeconstantC>0n2B(0;)andZZZr

8、11f(x)dx=dtfdS:B(0;)jxjn20@B(0;t)tn2RAs!0,I+J!1f(x)dx:Thus,B(0;)jxjn2Zr1Z11Zlim0(t)dt=f(x)dxfdy!0n(n2)(n)B(0;r)jxjn2rn2B(0