Coercive Bilinear form weakly lower-semicontinuous

Question

I'm trying to solve a nonlinear parabolic PDE. Let $(V,H,V^*)$ be a Gelfand-Triple and $a\colon V\times V \to \mathbb{R}$ a bounded bilinear form. We assume that $a$ is also coercive in the following sense. There exist $\alpha, c \geq 0$ with

\begin{align} \alpha \|v\|_V^2 \leq a(v,v) + c|v|_H^2. \end{align}

Is there any way to prove, that the linear form $a(v,v)$ is weakly lower-semicontinuous? It's quite easy to see, that, if $c=0$ holds, $a$ is weakly lower-semicontinuous. Namely, for $v_n \rightharpoonup v$

\begin{align} 0 \leq \alpha\|v_n - v\|_V^2 \leq a(v_n,v_n) - a(v,v_n) - a(v_n,v) + a(v,v) \end{align}

By applying $\liminf$ to this inequality, we get $a(v,v) \leq \liminf_{n\to\infty}a(v_n,v_n)$. Is there a similar proof if $c \neq 0$? Thank you in advance.

Answer 1

Your proof shows that the bilinear form $$ a_c(u,v) = a(u,v) + c(u,v)_H $$ is weakly lower semicontinuous. If the embedding $V\hookrightarrow H$ is compact, then also $a$ is weakly lower semi-continuous: Let $v_n\rightharpoonup v$ in $V$ and $v_n\to v$ in $H$. Then we get $$ \lim\inf a(v_n,v_n) = \lim\inf a_c(v_n,v_n) - \lim c\|v_n\|^2 \ge a_c(v,v) - c\|v_n\|_H^2 = a(v,v). $$