We prove the existence of generalized solutions of the Monge–Kantorovich equations with fractional ss-gradient constraint, 0<s<10<s<1, associated to a general, possibly degenerate, linear fractional operator of the type,
with integrable data, in the space
Λs,p0(Ω), which is the completion of the set of smooth functions with compact support in a bounded domain
Ω for the
Lp-norm of the distributional Riesz fractional gradient
Ds in
ℝd (when
s=1,
D1=D is the classical gradient). The transport densities arise as generalized Lagrange multipliers in the dual space of
L∞(ℝd) and are associated to the variational inequalities of the corresponding transport potentials under the constraint
|Dsu|≤g. Their existence is shown by approximating the variational inequality through a penalization of the constraint and nonlinear regularization of the linear operator
ℒsu. For this purpose, we also develop some relevant properties of the spaces
Λs,p0(Ω), including the limit case
p=∞ and the continuous embeddings
Λs,q0(Ω)⊂Λs,p0(Ω), for
1≤p≤q≤∞. We also show the localization of the nonlocal problems (
0<s<1), to the local limit problem with classical gradient constraint when
s→1, for which most results are also new for a general, possibly degenerate, partial differential operator
ℒ1u with coefficients only integrable and bounded gradient constraint.
