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Sci."],"published-print":{"date-parts":[[2024,8]]},"abstract":"<jats:p> We prove the existence of generalized solutions of the Monge\u2013Kantorovich 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. <\/jats:p>","DOI":"10.1142\/s1664360723500145","type":"journal-article","created":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T08:31:18Z","timestamp":1701160278000},"source":"Crossref","is-referenced-by-count":4,"title":["Nonlocal Lagrange multipliers and transport densities"],"prefix":"10.1142","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6284-8045","authenticated-orcid":false,"given":"Assis","family":"Azevedo","sequence":"first","affiliation":[{"name":"CMAT and Departamento de Matem\u00e1tica, Escola de Ci\u00eancias, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8438-0749","authenticated-orcid":false,"given":"Jos\u00e9 Francisco","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"CMAFcIO \u2013 Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade de Lisboa, P-1749-016 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0286-1616","authenticated-orcid":false,"given":"Lisa","family":"Santos","sequence":"additional","affiliation":[{"name":"CMAT and Departamento de Matem\u00e1tica, Escola de Ci\u00eancias, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2023,12,8]]},"reference":[{"key":"S1664360723500145BIB001","volume-title":"Sobolev Spaces","author":"Adams R. 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