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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 $|| α + β ||_{\infty} ≤ C$

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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.

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