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

$L^\alpha$-Regularization of the Beckmann Problem