Can someone help me for this problem?
Write the weak formulation of: $$\left\{\begin{align} -\frac{\partial^2u}{\partial x^2}-5\frac{\partial^2u}{\partial y^2}=f\quad&\text{in}\quad\Omega\subset\mathbb R^2\\ u=0 \quad&\text{in}\quad\partial\Omega \end{align}\right.$$ Then apply Lax-Milgram to show existence of weak solution.
Using Green identity I found the following weak formulation:
Find $u\in H_0^1(\Omega)$ solution of $$\int_\Omega u_x v_x+5\int_\Omega u_y v_y =\int_\Omega fv,\quad\forall\ v\in H_0^1(\Omega).$$
Is this correct? How can I now apply Lax-Milgram?
Can I transform this equation using gradient or Laplace operator? Thanks
You have to identify what are the bilinear form and the linear functional which correspond to your weak formulation.
In this case, the bilinear form is $B:H_0^1\times H_0^1\to\mathbb R$ defined by $$B[u,v]=\int_\Omega u_x v_x+5\int_\Omega u_y v_y.$$
And the linear functional is $\Lambda:H_0^1\to\mathbb R$ defined by $$\Lambda[v]=\int_\Omega fv.$$
Now, in order to prove existence of weak solution, you have to show that $B$ is continuous and coercive.