Steady state, one PDE

Considering groundwater flow with a pressure field \(P\) (i.e. head), a design parameter \(X\) and a general cost function

\[f(P(X),X)\]

\(P(X)\) solved by the Darcy / Richards equation

\[g(P(X),X) = 0\]

the optimization problem is given by

\[\begin{split}&\min_X f(C(P,X),X) \\ &s.t. h(C(P,X),X) = 0\end{split}\]

todo