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articleOpen Access
Nonlocal Lagrange multipliers and transport densities
Bulletin of Mathematical Sciences08 Dec 2023
Preview Abstract
We prove the existence of generalized solutions of the Monge–Kantorovich equations with fractional $s$ -gradient constraint, $0 < s < 1$ , associated to a general, possibly degenerate, linear fractional operator of the type,
$ℒ^{s} u = - D^{s} \cdot (A D^{s} u + b u) + d \cdot D^{s} u + c u,$
with integrable data, in the space $Λ_{0}^{s, p} (Ω)$ , which is the completion of the set of smooth functions with compact support in a bounded domain $Ω$ for the $L^{p}$ -norm of the distributional Riesz fractional gradient $D^{s}$ in $ℝ^{d}$ (when $s = 1$ , $D^{1} = D$ is the classical gradient). The transport densities arise as generalized Lagrange multipliers in the dual space of $L^{\infty} (ℝ^{d})$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $| D^{s} u | \leq g$ . Their existence is shown by approximating the variational inequality through a penalization of the constraint and nonlinear regularization of the linear operator $ℒ^{s} u$ . For this purpose, we also develop some relevant properties of the spaces $Λ_{0}^{s, p} (Ω)$ , including the limit case $p = \infty$ and the continuous embeddings $Λ_{0}^{s, q} (Ω) \subset Λ_{0}^{s, p} (Ω)$ , for $1 \leq p \leq q \leq \infty$ . We also show the localization of the nonlocal problems ( $0 < s < 1$ ), to the local limit problem with classical gradient constraint when $s \to 1$ , for which most results are also new for a general, possibly degenerate, partial differential operator $ℒ^{1} u$ with coefficients only integrable and bounded gradient constraint.