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Consider the following problem $$ \left\{ \begin{array} [c]{lll} -\Delta u+u_{x}+u=1 & ,\quad & x\in\left( 0,1\right) \times\left( 0,1\right) \\ u=0 & , & \Gamma_{1}=\left\{ 0\right\} \times\left[ 0,1\right] \text{ and }\Gamma_{2}=\left\{ 1\right\} \times\left[ 0,1\right] \\ \left\langle \nabla u,n\right\rangle =1 & , & \left[ 0,1\right] \times\left\{ 0\right\} \\ \left\langle \nabla u,n\right\rangle +u=3 & , & \left[ 0,1\right] \times\left\{ 1\right\} \end{array} \right. . $$ I used the notation $<,>$ for scalar product in $\mathbb{R}^n$ and $n$ is the unit normal vector. We need to prove that this problem admits a unique solution. I don't know who is the space of solution and how to define the bilinear form, linear form. There is a book we some examples like this one?

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You can start wrinting the variational formulation of the elliptic second order problem. Moving $u_{x}$ in $RHS$, multiplying by $$ v\in H_{\Gamma_{1},\Gamma_{2}}^{1}=\left\{ v\in H^{1}\mid v=0\text{ on }\Gamma_{1}\cup\Gamma_{2}\right\} $$ and then integrating we obtein (after applying a Green formula type) the following variational formulation $$ \text{Find }u\in H_{\Gamma_{1},\Gamma_{2}}^{1}\text{ such that }\forall v\in H^{1}:\int_{\Omega}\nabla u\cdot\nabla v+uv\operatorname*{dx}=\int_{\Omega }v-vu_{x}\operatorname*{dx}+\int_{\partial\Omega}v\left\langle \nabla u,n\right\rangle d\sigma, $$ where $\cdot$ is the scalar product and $u_{x}=\frac{\partial u}{\partial x}$.

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