Let $I$ be the open interval $(0, 2)$ and $V=H^1(I)$. Consider the bilinear form $$ a(u, v) = \int_0^2 u'v' + \left(\int_0^1 u\right) \left(\int_0^1 v\right) . $$
I'm trying to solve a problem in Brezis' Functional Analysis, i.e.,
Exercise 8.25
- Check that $a(u, v)$ is a continuous symmetric bilinear form, and that $a(u, u)=0$ implies $u=0$.
- Prove that $a$ is coercive. [Hint: Argue by contradiction and assume that there exists a sequence $\left(u_n\right)$ in $H^1(I)$ such that $a\left(u_n, u_n\right) \rightarrow 0$ and $\left\|u_n\right\|_{H^1}=1$. Let $\left(u_{n_k}\right)$ be a subsequence such that $u_{n_k}$ converges weakly in $H^1(I)$ and strongly in $L^2(I)$ to a limit $u$. Show that $u=0$.]
My below proof of (2.) is different from the suggested hint. Could you please check if I made some subtle mistakes?
We need an auxiliary result (in the same book)
Exercise 6.12 Let $X,Y,Z$ be real Banach spaces with corresponding norms $|\cdot|_X, |\cdot|_Y, |\cdot|_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection. Prove that for every $\varepsilon>0$ there is $C_\varepsilon > 0$ such that $$ |u|_Y \le \varepsilon |u|_X + C_\varepsilon |u|_Z \quad \forall u \in X. $$
Let $X := H^1 (I), Y := L^2 (I)$ and $Z:=\mathbb R$. Then $X \to Y, u \mapsto u$ and $Y \to Z, u \mapsto \int_0^1 u$ are compact linear operators. Fix $\varepsilon \in (0, 1/2)$. By Exercise 6.12, there is $C_\varepsilon > 0$ such that $$ \|u\|_{L^2} \le \varepsilon \|u\|_{H^1} + C_\varepsilon \left | \int_0^1 u \right | \quad \forall u \in H^1(I), $$ which implies $$ \|u\|^2_{L^2} \le 2\varepsilon [ \|u'\|_{L^2}^2 + \|u\|_{L^2}^2] + 2C_\varepsilon^2 \left | \int_0^1 u \right |^2 \quad \forall u \in H^1(I), $$ which implies $$ (1-2\varepsilon) [\|u'\|^2_{L^2} + \|u\|^2_{L^2}] \le (2C_\varepsilon^2+1) \left ( \|u'\|_{L^2}^2 + \left | \int_0^1 u \right |^2 \right ) \quad \forall u \in H^1(I), $$ which implies $$ (1-2\varepsilon) \|u\|^2_{H^1} \le (2C_\varepsilon^2+1) |a(u, u)|^2 \quad \forall u \in H^1(I). $$
