Show that form is coercive

Question

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with smooth boundary. Show that form $B:W^{1,2}_0(\Omega)\rightarrow \mathbb{R}$ such that $B(u,v)=\int_{\Omega}u_xv_x+2u_yv_y+u_xv$ is coercive.

Attempt:

We have $B(u,u)=\int_{\Omega}u_x^2+2u_y^2+u_xu\geqslant\int_{\Omega}|\nabla u|^2+\int_{\Omega}u_xu\geqslant C||u||_{W_0^{1,2}}^2+\int_{\Omega}u_xu$, where the last step is Poincare inequality. ($C=\min\{\frac{1}{2}, \frac{1}{2c}\}$)

Answer 1

Your argument is fine, except that you still need to deal with the term $\int_\Omega u \partial_x u$. To handle this note that since $u \in W^{1,2}_0$ we know that $u=0$ on $\partial \Omega$, and so we can compute $$ \int_\Omega u \partial_x u = \frac{1}{2} \int_\Omega \partial_x(u^2) = 0. $$ Hence $B(u,u) \ge C \Vert u\Vert_{W^{1,2}_0}^2$.