Coercivity - Weak Poisson's equation

Question

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 $\Vert \cdot \Vert_{1,2}$?

Answer 1

I figured it out. I can just make use of the equivalence of the norms: \begin{align} \begin{split} \vert u \vert_{1,2} \leq \Vert u \Vert_{1,2} &=\Bigl ( \int_{\Omega} u^2 \, dx + \int_{\Omega} \nabla u \cdot \nabla u \, dx \Bigr )^{\frac{1}{2}} \\ & \leq \Bigl ( c \int_{\Omega} \nabla u \cdot \nabla u \, dx+ \int_{\Omega} \nabla u \cdot \nabla u \, dx \Bigr )^{\frac{1}{2}} \\ &=\Bigl ( (c+1) \int_{\Omega} \nabla u \cdot \nabla u \, dx \Bigr )^{\frac{1}{2}} \\ &= \sqrt{c+1} \vert u \vert_{1,2}, \end{split} \end{align} where $c \in \mathbb R$ is the Poincare-constant.

Answer 2

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).