# A coupled problem of elliptic equations.

Let $$Ω$$ be an open bounded on class $$C^1$$. We consider, on $$Ω$$, the problem with the following limits:

$$(P)\quad \left\{ \begin{array}{lcc} -\Delta u_1 +\alpha u_2=f_1 & \text{on } \Omega, \\ \\ -\Delta u_2 +\beta u_1=f_2 & \text{on } \Omega, \\ \\ u_1 =u_2=0 & \text{on } \partial \Omega, \end{array} \right.$$

where $$\alpha,\beta\in L^{\infty}(\Omega)$$ and $$f_1,f_2\in L^{2}(\Omega)$$.

We note $$H=H_0^1(\Omega)\times H_0^1(\Omega)$$. If $$u\in H$$, then $$u=(u_1,u_2)$$, with $$u_1,u_2\in H_0^1(\Omega)$$, and

$$||u||^2_{H}=||u_1||^2_{H^1_0(\Omega)}+||u_2||^2_{H^1_0(\Omega)}$$.

1. Show, for an element $$u=(u_1,u_2)∈ H^2(Ω)×H^2(Ω)$$, the equivalence between the boundary problem $$(P)$$ and the variational problem $$(P_λ)$$, with $$λ> 0$$:

$$(P_λ)\quad \left\{ \begin{array}{lcc} u\in H \\ \\ \forall v\in H, \quad a_{λ}(u,v)=L_{λ}(v), \end{array} \right.$$,

where

$$a_{λ}(u,v)=\int_{\Omega}(∇u_1 · ∇v_1 + λ∇u_2 · ∇v_2) dx +\int_{\Omega}(αu_2v_1 + λβu_1v_2) dx$$,

$$L_{λ}(v)=\int_{\Omega}(f_1v_1 +\lambda f_2v_2)dx$$,

$$u=(u_1,u_2), v=(v_1,v_2)\in H$$

1. Suppose $$λ = 1$$. Show that there exists $$C> 0$$, depending only on $$Ω$$, such that if $$|| α + β ||_{\infty} ≤ C$$, then there is one and only one solution $$u\in H$$ of $$(P_{\lambda})$$

I did the variational formulation, but I do not know how I should make the $$\lambda$$ of the $$(P)$$ problem appear.

And for the part 2. I do not know how to relate the constant $$C$$ and the $$|| α + β ||_{\infty} ≤ C$$

Multiply the first equation $$-\Delta u_1 +\alpha u_2=f_1$$ by $$v_1$$ and the second equation $$-\Delta u_2 +\beta u_1=f_2$$ by $$\lambda v_2$$. Take the sum, integrate over $$\Omega$$ and then integrate by parts.