Problem Formulation
1. Variational formulation
Let \(p\) and \(q\) the slopes according respectively to the \(x\) and the \(y\) axis. For each test function \(v\), the solution of the following problem give us the height \(z\) from the surface we want to rebuild :
\[\int_\Omega \Delta z v = \int_\Omega \nabla\cdot((p,q))v\]
By the use of Green formula on the left member :
\[\int_\Omega \Delta z v = -\int_\Omega \nabla z \cdot \nabla v + \int_{\partial\Omega}\frac{\partial z}{\partial n} v\]
And the same on the right member :
\[\int_\Omega \nabla\cdot ((p,q)) v = -\int_\Omega(p,q)\cdot\nabla v + \int_{\partial\Omega}(p,q)\cdot\vec{n}v\]
We have then :
\[-\int_\Omega \nabla z \cdot \nabla v + \int_{\partial\Omega}\frac{\partial z}{\partial n} v =-\int_\Omega(p,q)\cdot\nabla v + \int_{\partial\Omega}(p,q)\cdot\vec{n}v\]
We can notice that :
\[\int_{\partial\Omega}\frac{\partial z}{\partial n} v = \int_{\partial\Omega}(p,q)\cdot\vec{n}v\]
Finally, we obtain the weak formula :
\[\int_\Omega \nabla z \cdot \nabla v = \int_\Omega(p,q)\cdot\nabla v\]
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The previous problem has a solution up to a constant, to ensure uniqueness we select the zero-mean solution, i.e
\[\int_\Omega z = 0\]
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