Given the weak formulation of the Poisson equation, i.e. For given source function $f\in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such that
$$\int_{\Omega}\nabla u \cdot \nabla v \, dx= \int_{\Omega}fv \, dx \quad \text{for all } \,\,v\in H_0^1(\Omega).$$
To then apply Lax-Milgram one needs $a(u,v)=\int_{\Omega}\nabla u \cdot \nabla v \, dx$ to be coercive. So what I get is
$$a(u,u)=(\nabla u, \nabla u)_{L^2}=\Vert \nabla u \Vert_{L^2}^2=\vert u\vert_{1,2}^2$$
particularly $$a(u,u) \geq c \vert u\vert_{1,2}^2 $$
where $c=1$. Is the semi-norm sufficient or do I need to show it with regard to the norm $\Vert \cdot \Vert_{1,2}$?
Best Answer
You need to use the norm, but you can just use Poincaré's inequality (that in fact shows that the semi-norm is in this case an actual norm).