I have to show the following: Let $U\subset \mathbb{R}^n$ be nice (i.e. bounded, open and boundary of class $C^1$). Further there's a function
$f:H^1(U) \to \mathbb{R}^n$
which is continuous and satisfies: $r\in \mathbb{R} \wedge f(r\mathbf1) = 0\Rightarrow r=0$ and $\forall \lambda\ge 0, u\in H^1(U): f(\lambda u)=\lambda f(u)$. Now I want to show a certain type of a Poincaré inequality, i.e.
$\|u\|_{L^2}^2 \le C(U,f)(\|\nabla u\|^2_{L^2}+|f(u)|^2) \hspace{12pt} \forall u \in H^1(U)$
The hint was: Prove by contradiction and use Rellich-Kondrachov embedding theorem.
First of all, I know the proof for a Poincaré type inequality for a closed subspace of $H^1$ which does not contain the non zero constant functions.
This is what I did so far:
Suppose not, then there are $c_k \to \infty$ such that $0\not= u_k\in H^1(U)$ with
$ c_k(\|\nabla u_k\|^2_2+|f(u_k)|^2) < \|u_k\|^2_2 $
As in the proof of the above mentioned theorem, I assume $\|u_k\|_2=1$. Now we have
$ c_k(\|\nabla u_k\|_2^2 +|f(u_k)|^2)<1$
Since the term in the bracket is positive, we have $ (\|\nabla u_k\|_2^2 +|f(u_k)|^2)\to 0$ as $c_k\to\infty$, now my first question:
Can I deduce from here, that $(\|\nabla u_k\|_2^2 +|f(u_k)|^2)$ must be bounded, therefore $u_k$'s would be bounded in $H^1$. Is this obvious or how could I show that?
Hence due to Rellich-Kondrachov embedding theorem, there's a subsequence $u_{k_l}$ which is Cauchy in $L^2$ and therefore converges in $L^2$ to $u\in L^2$. Since $\|u_k\|_2=1$ we have $u\not=0$.
Now I should show that from $(\|\nabla u_k\|_2^2 +|f(u_k)|^2)\to 0$ it follows that $u_k \to 0$. This would be my contradiction. I guess here I have to use the property of $f$, since I have them not used yet. A hint would be much appreciated.
There's a second part of this exercises and unfortunately I do not know how to start:
$ H_\Gamma:=\{u\in H^1(U);u=0 \mbox{ on } \Gamma\subset \partial U\}$
and the volume of $\Gamma$ with respect to the surface measure is strictly positive. Then there's a constant $C>0$ not depending on $u$ such that:
$\|u\|_{H^1}\le C \|\nabla u\|_2$
Thanks a lot for your help.
math