Let be the unit disk in and let denote the boundary. Show that the only solution of the boundary value problem:
in
on
is . The hint given is to apply the 2-D divergence theorem on the vector field , but I don't see how that even enters in.
Let be the unit disk in and let denote the boundary. Show that the only solution of the boundary value problem:
in
on
is . The hint given is to apply the 2-D divergence theorem on the vector field , but I don't see how that even enters in.
You are going to want to calculate an integral two different ways
First note that
The right hand side of the above equation is non negative so we get that
Now consider the flux integral out of the disk
This is zero because u=0 on the boundary.
Now if we use the divergence theorem we get
This gives
Combining these two equations gives
So the integrand must be equal to zero so
If it is the first cases we are done if it is the plug that into the pde and conclue.