Really Struggling :(
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).
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?
Originally Posted by OliviaB
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
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.