Proving coercivity for weak form

[Carla1985]

I have the weak form of the poisson equation as $a(u,v)=l(v)$ where
$$a(u,v)=\int_\Omega \bigtriangledown u\cdot\bigtriangledown v$$

I have been proving existence and uniqueness of the solution using the Lax-Milgram Thm. I am stuck on proving that the bilinear form $$a(u,v)$$ is coercive. I have that it is coercive if there exists a constant c such that
$$|a(u,v)|\leq c||v||_V\ \ \forall\ v\in V$$
where $||\cdot||_V$ is the norm corresponding to scalar product $(\cdot,\cdot)_V$.

I think I need to use the Poincare inequality to solve it but not sure how. Could someone please give me tips on how to go about proving this please? Thanks

 

[Euge]

Hi Carla1985,

There are pieces of information that are missing in your question. What are the conditions on $\Omega$? In what space do $u$ and $v$ belong to? Also, in the definition of coercivity, you should have $|a(v,v)| \ge c\|v\|^2$ for all $v\in V$.

 

[Carla1985]

Hi Carla1985,

There are pieces of information that are missing in your question. What are the conditions on $\Omega$? In what space do $u$ and $v$ belong to?

I have my spaces as:
$$u\in U:=\{u\in H^1(\Omega)\ |\ u=g_D\textsf{ on }\partial\Omega_D\}$$
$$v\in V:=\{v\in H^1(\Omega)\ |\ v=0\textsf{ on }\partial\Omega_D\}$$

Also, in the definition of coercivity, you should have $|a(v,v)| \ge c\|v\|^2$ for all $v\in V$.

Yes sorry, that's what I should have typed :)

 

[Euge]

The coercivity of $a$ follows directly from the Poincare inequality, for there is a constant $C = C(\Omega)$ such that $a(u,u) = \|\nabla u\|_{L^2(\Omega)}^2 \ge C\|u\|^2$ for all $u\in H^1(\Omega)$.