  We investigate the problem of optimal transport in the so-called Beckmann
form, i.e. given two Radon measures on a compact set, we seek an optimal flow
field which is a vector valued Radon measure on the same set that describes a
flow between these two measures and minimizes a certain linear cost function.
  We consider $L^\alpha$ regularization of the problem, which guarantees
uniqueness and forces the solution to be an integrable function rather than a
Radon measure. This regularization naturally gives rise to a semi-smooth Newton
scheme that can be used to solve the problem numerically. Besides motivating
and developing the numerical scheme, we also include approximation results for
vanishing regularization in the continuous setting.
