Nonlocal Lagrange multipliers and transport densities

Abstract

We prove the existence of generalized solutions of the Monge–Kantorovich equations with fractional [Formula: see text]-gradient constraint, [Formula: see text], associated to a general, possibly degenerate, linear fractional operator of the type, [Formula: see text] with integrable data, in the space [Formula: see text], which is the completion of the set of smooth functions with compact support in a bounded domain [Formula: see text] for the [Formula: see text]-norm of the distributional Riesz fractional gradient [Formula: see text] in [Formula: see text] (when [Formula: see text], [Formula: see text] is the classical gradient). The transport densities arise as generalized Lagrange multipliers in the dual space of [Formula: see text] and are associated to the variational inequalities of the corresponding transport potentials under the constraint [Formula: see text]. Their existence is shown by approximating the variational inequality through a penalization of the constraint and nonlinear regularization of the linear operator [Formula: see text]. For this purpose, we also develop some relevant properties of the spaces [Formula: see text], including the limit case [Formula: see text] and the continuous embeddings [Formula: see text], for [Formula: see text]. We also show the localization of the nonlocal problems ([Formula: see text]), to the local limit problem with classical gradient constraint when [Formula: see text], for which most results are also new for a general, possibly degenerate, partial differential operator [Formula: see text] with coefficients only integrable and bounded gradient constraint.