{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:27:02Z","timestamp":1787318822263,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"name":"Consejeria de Transformacion Economica, Industria, Conocimiento y Universidades of the Regional Government of Andalusia","award":["UAL2020-FQM-B2046"],"award-info":[{"award-number":["UAL2020-FQM-B2046"]}]},{"name":"Institut de Recherche Mathematique de Rennes"},{"DOI":"10.13039\/501100002924","name":"Federaci\u00f3n Espa\u00f1ola de Enfermedades Raras","doi-asserted-by":"publisher","award":["PID2020-116809GB-I00"],"award-info":[{"award-number":["PID2020-116809GB-I00"]}],"id":[{"id":"10.13039\/501100002924","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2024,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The paper is devoted to divergence-curl results involving a divergence free measure-valued field [Formula: see text], where [Formula: see text] is a signed Radon measure on [Formula: see text] and [Formula: see text] is a nonvanishing regular vector field in [Formula: see text], and a gradient measure-valued field [Formula: see text] on [Formula: see text], [Formula: see text]. On the one hand, in a nonperiodic framework we prove that for any open set [Formula: see text] of [Formula: see text], the orthogonality condition [Formula: see text] in [Formula: see text] implies the equality [Formula: see text] in [Formula: see text]. The key ingredient of the proof is based on the existence of a representative in [Formula: see text] of the bounded variation function [Formula: see text] in [Formula: see text]. This result allows us to extend in the setting of ODE\u2019s flows the famous Franks\u2013Misiurewicz theorem, which claims that the Herman rotation set of any continuous two-dimensional flow on the torus [Formula: see text] is a closed line segment of a line of [Formula: see text] passing through [Formula: see text]. Moreover, this nonperiodic divergence-curl result can be applied to a finite almost periodic bounded variation function [Formula: see text] and to a finite almost periodic measure-valued field [Formula: see text]. On the other hand, in the periodic case with dimension [Formula: see text], assuming that [Formula: see text] is absolutely continuous with respect to Lebesgue\u2019s measure on the torus [Formula: see text], we prove that if the product [Formula: see text] is the zero measure on [Formula: see text], so is the product of the [Formula: see text]-means [Formula: see text].<\/jats:p>","DOI":"10.1137\/23m1617539","type":"journal-article","created":{"date-parts":[[2024,9,17]],"date-time":"2024-09-17T12:40:38Z","timestamp":1726576838000},"page":"6398-6421","source":"Crossref","is-referenced-by-count":0,"title":["A New Divergence-Curl Result for Measures. 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