Use Lax-Milgram theorem to prove the existence of weak solution for an elliptic equation Let $\Omega$ an open bounded and regular domain to $\mathbb{R^n}$ and let $\{\overline{\Omega_1},\overline{\Omega_2}\}$ a partition of $\Omega.$ 
$\bar{\Omega} = \bar{\Omega_1} \cup \bar{\Omega_2}.$ We put $\Gamma = \partial \Omega_1 \cap \partial \Omega_2$ the interface between $\Omega_1$ and $\Omega_2$ such that $\Gamma \subset  \Omega.$
We put $u_1 = u/\Omega_1$ (restriction $u$ to $\Omega_1$) and we condider the problem:
$$-k_i \Delta u_i = f , x \in \Omega_1 , i=1,2$$
$$u_1=0 , x \in \partial \Omega$$
$$u_1 = u_2 , x \in \Gamma$$
$$k_1 \nabla u_1 n = k_2 \nabla u_2 n , x \in \Gamma$$
$k(x)$ is an piecewise constant , with $k(x) = k_i > 0, i = 1,2$  and $f \in L^2(\Omega)$
My questions are, how:
1-prouve that this problem admit a unique solution in an adequat Hilbert space $V.$
2- Prouve that $\exists c > 0, ||u||_V \leq c$ and prouve that $u$ verfies the minimum of energy.
 A: The big picture is: If $u$ satisfies those two continuity conditions on $\Gamma$ together with the equation in each subdomain, then $u$ solves the weak problem for the following diffusion equation on the whole domain:
$$\left\{\begin{aligned}
-\nabla \cdot (k\nabla u )&= f \quad\text{in } \Omega,
\\
u&=0 \quad\text{on } \partial \Omega.
\end{aligned}\right.
$$ 

Multiply both equations by a same $H^1_0(\Omega)$ test function $v$:
$$
-\int_{\Omega_1}k_1 \Delta u_1\,v -\int_{\Omega_2}k_2 \Delta u_2\,v= \int_{\Omega}f\,v.
$$
Integrating by parts using Green's identity on each domain:
$$
\int_{\Omega_1}k_1 \nabla u_1\cdot\nabla v - \int_{\partial \Omega_1}k_1 (\nabla u_1\cdot n_1)v  + \int_{\Omega_2}k_2 \nabla u_2\cdot\nabla v - \int_{\partial \Omega_2}k_2 (\nabla u_2\cdot n_2)v =  \int_{\Omega}f\,v,
$$
where $n_1$ and $n_2$ are the unit vector normal to the boundaries of $\Omega_1$ and $\Omega_2$ respectively. Now we focus on the boundary part, for $\partial \Omega_1\cap \partial \Omega_2 = \Gamma$:
$$
\begin{aligned}
&\int_{\partial \Omega_1}k_1 (\nabla u_1\cdot n_1)v + \int_{\partial \Omega_2}k_2 (\nabla u_2\cdot n_2)v 
\\
=& \int_{\partial \Omega_1\backslash\Gamma}k_1 (\nabla u_1\cdot n_1)v + \int_{\partial \Omega_2\backslash\Gamma}k_2 (\nabla u_2\cdot n_2)v + \int_{\Gamma} (k_1 \nabla u_1\cdot n_1 + k_2\nabla u_2\cdot n_2)v
\\
=& \int_{\partial \Omega}k (\nabla u\cdot n)v + \int_{\Gamma} (k_1 \nabla u_1\cdot n_1 + k_2\nabla u_2\cdot n_2)v.
\end{aligned}
$$ 
The first term vanishes because $v=0$ on $\partial \Omega$. Second term vanishes because $$k_1 \nabla u_1\cdot n - k_2\nabla u_2\cdot n = k_1 \nabla u_1\cdot n_1 + k_2\nabla u_2\cdot n_2 = 0,$$
by $n_1$ and $n_2$ share the same magnitude but point the opposite direction pointwisely. By the other interface condition $u_1 = u_2$ on $\Gamma$, we know $u\in H^1_0(\Omega)$, therefore
$$
\int_{\Omega} k\nabla u\cdot \nabla v = \int_{\Omega}f\,v,\quad \forall v\in H^1_0.\tag{1}
$$
To prove the existence of a solution in $V = H^1_0$, we need to fulfill the conditions of Lax-Milgram theorem:


*

*$\displaystyle \int_{\Omega} k\nabla u\cdot \nabla v \leq C\|u\|_{H^1_0(\Omega)} \|v\|_{H^1_0(\Omega)}$, this is trivial by Cauchy-Schwarz inequality.

*Coercivity relies on Poincaré inequality for $v\in H^1_0$: $\|v\|_{L^2(\Omega)} \leq c\|\nabla v\|_{L^2(\Omega)}$, so we have 
$$\int_{\Omega} k|\nabla v|^2 \geq \alpha (\|v\|_{L^2(\Omega)}^2 + \|\nabla v\|_{L^2(\Omega)}^2)$$
for some constant $\alpha>0$ (You need to work out this $\alpha$, it depends on $k_1$, $k_2$ and $c$ in Poincaré inequality).
The second question is then trivial, let $v=u$ in (1), using Cauchy-Schwarz and Poincaré inequality on the right hand side:
$$
\alpha \|u\|_{H^1_0(\Omega)}^2\leq  \int_{\Omega} k|\nabla u|^2 =  \int_{\Omega}f\,u\leq \|f\|_{L^2(\Omega)} \|u\|_{L^2(\Omega)} \leq c\|f\|_{L^2(\Omega)} \|u\|_{H^1_0(\Omega)},
$$ 
hence the Céa's lemma holds:
$$\|u\|_{H^1_0(\Omega)} \leq \frac{c}{\alpha}\|f\|_{L^2(\Omega)}.$$
And if $u$ solves (1), $u$ also minimize the energy functional:
$$
\mathcal{F}(u) = \frac{1}{2}\int_{\Omega} k|\nabla u|^2 - \int_{\Omega}f\,u.
$$

There are several typos in your question, please double check your notes:


*

*It should be $u_1 = u|_{\Omega_1}$.

*It should be $u= 0$ on $\partial \Omega$, not just $u_1$.

*There should be a dot product between the gradient and the normal, $k_1 \nabla u_1\cdot n= k_2\nabla u_2\cdot n$ because they are vectors.

*"prove", not "prouve".

*Céa's lemma has some norm of $f$ on the right hand side, because $\|u\|_V$ depends on that, you can't find a single $c$ satisfying that property.

