Weak formulation of Poisson’s equation: is the bilinear form coercive

Question

Let us consider the Poisson's equation
$$-\nabla ^{2}u=f,$$
on a domain $\Omega \subset {\mathbb R}^{d}$ with $u=0$ on its boundary. Use the $L^{2}$-scalar product
$$\langle u,v\rangle =\int _{\Omega }uv\,dx$$
to derive the weak formulation. Then, testing with differentiable functions $v$ yields
$$ -\int _{\Omega }(\nabla ^{2}u)v\,dx=\int _{\Omega }fv\,dx.$$
The left side of this equation can be simplified by integration by parts using Green's identity and assuming that $v=0$ on $\partial\Omega$:
$$ \int _{\Omega }\nabla u\cdot \nabla v\,dx=\int _{\Omega }fv\,dx.$$
The generic form is obtained by assigning
$$a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v\,dx$$
and
$$f(v)=\int _{\Omega }fv\,dx.$$
i.e. $a(u,v)=f(v)$. Is the bilinear form coercive? If yes, how could I prove it?

Answer 1

The coercivity depends on the topology you put on your function space. In this setting we can take as a function space $H^1_0(\Omega)$, with norm $$ \|u\|_{H^1_0(\Omega)}=\|u\|_{L^2}+\|\nabla u\|_{L^2}. $$ It is defined as the closure in $H^1(\Omega)$ of the space of test functions $\mathscr D(\Omega)$ under the above norm.

The quadratic form associated to your bilinear form $a$ is exactly $$ a(u,u)=\|\nabla u\|_{L^2}^2, $$ so one only needs to bound the remaining quantity $\|u\|_{L^2}$.

At this point, if your domain is bounded, you can use Poincare’s inequality: $$ \|u\|_{L^2}\leq C \|\nabla u\|_{L^2} \text{ for any } u\in H^1_0(\Omega). $$

With this estimate you then have coercivity: $$ a(u,u)\geq \frac{1}{4C^2}\|u\|_{H^1_0(\Omega)}^2. $$

This argument actually shows that you can use $\|\nabla u\|_{L^2}$ as an equivalent norm for the space $H^1_0$.

Edit.

You can still do the same thing if your domain is unbounded but such that its image under one of the coordinates is bounded (in this case Poincare’s inequality still holds). If the domain is unbounded but $d\geq 3$, you can still do something using Sobolev embeddings (I could try to find a reference if you are interested).