Laplacian and Fredholm Alternative
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.