I haven't made any progress on this one, but this is Poisson's equation, isn't it? And its inhomogenous? I think I need to use Greens identity to solve this. Is that right?
Hi,
I am really struggling with a subject at my university which is all about partial differential equations. It takes me a long time to understand even the most basic concepts, and I really need some help understanding and completing the following questions.
Let be a bounded region in , and suppose on .
(i) If is a solution of
show that on
(ii) If is a solution of
show that is unique (It can be assumed that part (i) is true).
Well, it looks like a typical application of the Lax-Milgram theorem to me.
The weak formulation of this problem on reads
or, by integrating the left hand side by parts and using the boundary conditions,
.
Define the bilinear form on ,
.
To show that is solvable for some , we can invoke the Lax-Milgram theorem: We need just show that is bounded and coercive.
You can show it is bounded, ie there exists a such that
, where
As for being coercive, we must show there is with . We compute
.
Now the Lax-Milgram theorem applies, and there is a such that
.
And reflexivity implies .
For part ii), consider two solutions of the problem at hand.
Then their difference is a solution for the problem given in part i), and therefore must be zero.
Ps. You cannot construct a Green's function here, as zero is an eigenvalue for on a region.