lax milgram on a problem We have the following problem:\
$
\begin{array}{cc}
  \{ & 
    \begin{array}{cc}
      -\Delta(u)+u=1 & ,\Omega=(0,1)\times(0,1) \\
      u=0 & ,\Gamma_{1}=[0,1]\times\{0\}, \Gamma_{2}=\{0\}\times[0,1]\\
      \frac{\partial u}{\partial n}=1 & ,\Gamma_{4}=\{1\} \times [0,1]\\
      \frac{\partial u}{\partial n}+u=1 &, \Gamma_{3}=[0,1]\times \{1\}
    \end{array}
\end{array}
$
I was asked to write the variational form and to prove the existence and the uniqness of the solution;
My idea is to apply Lax Milgram theorem. I wrote the variational form, which is the following:
$\int_{\Omega} \nabla u \nabla v +\int_{\Omega}uv +\int_{\Gamma_{1}}\frac{\partial u}{\partial y}v+\int_{\Gamma_{2}}\frac{\partial u}{\partial x}v-\int_{\Gamma_{3}}\frac{\partial u}{\partial y}v=\int_{\Omega}v+\int_{\Gamma_{4}}v$, $\forall v\in H^{1}_{0}$
I thought that I may take $a(u,v)$ to be the left side member and $l(v)$ be the right side member. Now I am completely stuck at the part when I have to prove that $a$ is bounded and coercitive. I proved that $a$ is bilinear and so is $l$, but the rest I dont know. Could you give me any suggestion, please? Thank you very much! I am stuck beacuse I dont know how to handle that integrals on that parts of the boundary of $\Omega$..
 A: Firts of all the weak formulation of your Poisson problem is:
Let $v$ be test function in a space which is later on determined. Let integrate the second order term by parts and then:
$$\int_{\Omega}{\left[\vec{grad}{\,u}\cdot\vec{grad}{\,v}+uv\right]\,dV}=\int_{\partial \Omega}{v\frac{\partial u}{\partial n}d\sigma}+\int_{\Omega}{v\,dV}$$ 
Now, apply the boundary conditions on $\partial \Omega=\Gamma_1\cup\Gamma_2\cup \Gamma_3\cup\Gamma_4$ imposing that $v=0$ in $\Gamma_1\cup\Gamma_2$ to obtain:
$$\int_{\Omega}{\left[\vec{grad}{\,u}\cdot\vec{grad}{\,v}+uv\right]\,dV}+\int_{\Gamma_3}{uv\,d\sigma}=\int_{\Omega}{v\,dV}+\int_{\Gamma_4}{v\,d\sigma}$$
Now it suffices for  $v$ to belong to $H^{1}(\Omega)$ space with $v=0$ in $\Gamma_1\cup\Gamma_2$ and $u\in H^1(\Omega)$
For  the sake of clarity we name:
$$a(u,v)=\int_{\Omega}{\left[\vec{grad}{\,u}\cdot\vec{grad}{\,v}+uv\right]\,dV}+\int_{\Gamma_3}{uv\,d\sigma}$$
and
$$l(v)=\int_{\Omega}{v\,dV}+\int_{\Gamma_4}{v\,d\sigma}$$
To prove that the solution of the Poisson problem exists and it is unique, the bilinear form $a(u,v)$ has to be contiuous and be coercive and $l(v)$ has to be continuous in $\Omega$ (Lax-Milgram)
The norm of $a(u,v)$ can be bounded using the triangle inequality as follows:
$$|a(u,v)|\leq \bigg|\int_{\Omega}{\left[\vec{grad}{\,u}\cdot\vec{grad}{\,v}+uv\right]\,dV}\bigg|+\bigg|\int_{\Gamma_3}{uv\,d\sigma}\bigg|$$
applying Cauchy-Schwarz inequality under the notation:
$$||\cdot||_{0,\Omega}=||\cdot||_{L^2(\Omega)} \qquad ||\cdot||_{1,\Omega}=||\cdot||_{H^1(\Omega)}  $$
one obtains
$$|a(u,v)|\leq  ||\vec{grad}{\,u}||_{0,\Omega}||\vec{grad}{\,v}||_{0,\Omega}+||u||_{0,\Omega}||v||_{0,\Omega}+||u||_{0,\Gamma_3}||v||_{0,\Gamma_3}$$
As we know $||\cdot||_{0,\Omega}\leq||\cdot||_{1,\Omega}$ and $||\cdot||_{0,\Gamma_3}\leq ||\cdot||_{0,\Omega}\leq||\cdot||_{1,\Omega}$ and therefore it can be proved that:
$$|a(u,v)|\leq C_1 ||u||_{1,\Omega}||v||_{1,\Omega}\tag{*}$$
with $C_1$ strictly positive, i.e. $C_1>0$. As a result the bilinear form $a(u,v)$ is continuous in $\Omega$.
If we have $a(u,u)$ it can be proved that:
$$|a(u,v)|=\bigg| \int_{\Omega}{\left[|\vec{grad}{\,u}|^2+u^2\right]\,dV}+\int_{\Gamma_3}{u^2\,d\sigma}\bigg|=\bigg|||u||^2_{1,\Omega} + ||u||^2_{0,\Gamma_3}\bigg|\leq C_2||u||^2_{1,\Omega}\tag{**}$$
with $C_2>0$ and therefore we can assert that the bilinear form $a(u,v)$ is coercive.
Finally let us prove that $l(v)$ is continuous applying triangle inequality and  Cauchy-Schwarz inequality
$$|l(v)|=\bigg| \int_{\Omega}{v\,dV}+\int_{\Gamma_4}{v\,d\sigma}\bigg|\leq \bigg|\int_{\Omega}{v\,dV}\bigg|+\bigg|\int_{\Gamma_4}{v\,d\sigma}\bigg|\leq C_{*}(\Omega)||v||_{0,\Omega} + C_{**}(\Gamma_4)||v||_{0,\Gamma_4}$$
Recall that  $||\cdot||_{0,\Omega}\leq||\cdot||_{1,\Omega}$ and $||\cdot||_{0,\Gamma_4}\leq ||\cdot||_{0,\Omega}\leq||\cdot||_{1,\Omega}$ and then 
$$|l(v)|\leq C_5 ||v||_{1,\Omega} \tag{***}$$
with $C_5>0$. Proving that $l(v)$  is continuous in $\Omega$.
From $(*)$, $(**)$ and $(***)$ we have proved the existence and uniqueness of your Poisson problem.
Hope this helps
