Primal Dual model in the continuous domain The continuous max flow problem is posed as follows : 

sup $\int_\Omega  p_s(x)dx$

subject to : 

$|p(x)| \le C(x);  \forall x \in \Omega $

$p_s(x) \le C_s(x);  \forall x \in \Omega $

$p_t(x) \le C_t(x);  \forall x \in \Omega $

$\nabla \cdot p(x) - p_s(x) + p_t(x) = 0;  \forall x \in \Omega $

Here $p(x)$ is a field vector and is analogous to the flow in the discrete domain. $\nabla \cdot p$ is the divergence of the field p.

How do i find out the dual of this maximization problem using the lagrangian dual technique, i.e. the equivalent min cut formulation of the problem in the continuous domain.
Your question may be answered in the following paper:

@article{strang1983maximal,
  title={Maximal flow through a domain},
  author={Strang, G.},
  journal={Mathematical Programming},
  volume={26},
  number={2},
  pages={123--143},
  year={1983},
  publisher={Springer}
}