Apart from the classic Lax-Milgram theorem which is powerful under the Hilbert setting, there are two generalizations: (from Wikipedia)
- Babuska-Lax-Milgram theorem
suppose $U,V$ Hilbert spaces (real-valued), $B: U\times V\rightarrow \mathbb{R}$ a continuous bilinear form, if $B$ is weakly coercive:
for some constant $c>0$, $\forall u\in U$,
$$
\sup_{||v||=1}B(u,v)\geq c||u||
$$
and for all $0\neq v\in V$
$$
\sup_{||u||=1}|B(u,v)|>0
$$
then for $f\in V^\ast$, there exists an unique solution $u_f\in U$ to the weak formulation:
$$
B(u_f,v)=(f,v),\quad \forall v\in V
$$
moreover, the solution depends continuously on the datum:
$$
||u_f||\leq\frac{1}{c}||f||
$$
- Lions-Lax-Milgram theorem
Let $H$ a Hilbert space and $V$ a normed space, $B:H\times V\rightarrow \mathbb{R}$ a continuous bilinear form, the following are equivalent:
for constant $c>0$,
$$
\inf_{||v||_V=1}\sup_{||h||_H\leq 1}|B(h,v)|\geq c;
$$
for each continuous linear functional $f\in V^\ast$, there exists $h\in H$ s.t.
$$
B(h,v)=(f,v)\quad \forall v\in V.
$$
I am thinking are there any specific reason or problem to apply these two theorem? In Wikipedia, it says the second theorem may be useful for problem with time-dependent boundary where the classic Lax-Milgram cannot apply, but it does not provide more justification like how the second one can prove the existence.
Does someone see these two theorem applied in any specific problem? It would be great if you can offer some links or reading list for me.
Best Answer
I have never seen the first theorem, but googling for it I have found an entry of the encyclopedia of maths that refers to a 1972 survey article by Babuska in which some applications to numerical analysis are seemingly given.
Concerning the second one, you can take a look at Chapter 14 of this Internet course.