I suddenly think that I should post this entry ‘cos sometimes I don’t remember these stuffs. These results appear frequently in solving PDEs especially when using the direct method. For example, the simplest case is the following eigenvalue problem
over a bounded domain with Dirichlet boundary condition. We assume . Our aim is to show the existence of the first eigenvalue . Obviously, our problem is to solve the following optimization
The direct method says that we firstly select a minimizing sequence, say , then we need to prove is convergent. There are two steps in the above argument which lead to this entry. Our first claim is the following.
Boundedness implies weakly convergence. The first result says that
If is a reflexive Banach space and is a bounded sequence. Then up to a subsequence converges weakly to some in .
The proof of this claim can be found in a book due to Brezis (Theorem III.27). Interestingly, its converse also holds by the Eberlein-Šmulian theorem.
Theorem (Eberlein-Šmulian). Suppose is a Banach space such that every bounded sequence contains a weakly convergent subsequence. Then is reflexive.
There was an elementary proof of this theorem. We refer the reader to a paper due to Whitley [here]. Let us get back to our optimization problem. Once we have a minimizing sequence it is clear to see that is bounded in since
By using the first claim, converges weakly to some . It is worth noticing that by saying in we mean in and in . Now we need further argument
Weakly convergence becomes strongly convergence via compact operator. This second result says that
A compact operator between Banach spaces maps every weakly convergent sequence in into one that converges strongly in .
The proof of this relies on the contradiction argument and the fact that once a sequence converges strongly to some limit, this limit is unique. However, the converse is no long true. For example, by the Schur theorem, a sequence in converges weakly, it also converges strongly. We take to be the identity in . Since has infinity dimensional, by using the Riesz theorem, cannot be compact.
Using this claim we deduce that converges strongly to in for any . By using the Minkowski and Holder inequalities we can show that satisfies the constraint. It now follows from the weakly lower semi-continuous of norm that indeed satisfies the equation. The proof follows.