Here is the question:
"Consider the boundary value problem in lR^2:
Laplacian(u)=(1/r)*d/dr(r*du/dr)=c .for r^2=x^2+y^2<1
du/dr=2 .on r=1
Show there is no solution unless c=4, and find solutions in this case."
I think I am supposed to use the Fredholm alternative:
L(u) = f has a solution if and only if <f,v>=0 for all v in ker(L*) where L* is the adjoint.
My problem is basically finding the boundary conditions for L*. Since <u,v> involves the integral between 0 and 1, and so the 1/r is infinity when r=0.
I get my adjoint problem to be:
You can see the third term doesn't really work when I stick in r=0.
Any help would be much appreciated.
I'm not confident that I know what I'm talking about here, but it seems to me that if you're looking at a Laplacian in polar coordinates then you should be using the polar version of areal measure, in other words rdrdθ:
That has the big advantage of killing off the embarrassing factor 1/r, because it gets multiplied by r.
Since θ does not figure explicitly in the calculation, we can ignore it and just do the r integration by parts. I get the answer to be that L* is the same differential operator as L, with boundary condition 2v(1) – u(1)v'(1) = 0.
Cheers for looking at that Opalg...
think you have spotted my mistake... didn't bother actually checking my adjoint worked, but rather just used a formula for what it should be.
Obviously the formula only works when it is d/dx.
I guess this is why you should understand how formulas work before you use them!