{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T01:49:55Z","timestamp":1787363395495,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In this paper we study the existence of solutions for a type of nonlocal minimization problem. The minimization principle is relaxed by means of Young measures, and different generalized necessary conditions of optimality are used: the Euler\u2013Lagrange equation and the extended version of the Weierstrass minimum principle. These conditions are applied to establish some results concerning the existence and the description of the simplest relaxation of the problem in terms of Young measures. The main result of the paper is the description of the simplest relaxation by considering only a combination of two Dirac measures. The research is focused on the homogeneous and nonhomogeneous 1-dimensional scalar cases. The results obtained are numerically exploited to compute the lower semicontinuous envelope for some academic examples. A detailed numerical analysis of the nonlocal regularization for Young's tacking problem formulated in [D. Brandon and R. Rogers, Appl. Math. Optim., 25 (1992), pp. 287\u2013301] is given.<\/jats:p>","DOI":"10.1137\/080740635","type":"journal-article","created":{"date-parts":[[2010,4,7]],"date-time":"2010-04-07T18:08:11Z","timestamp":1270663691000},"page":"3838-3858","source":"Crossref","is-referenced-by-count":3,"title":["Some Aspects of the Existence of Minimizers for a Nonlocal Integral Functional in Dimension One"],"prefix":"10.1137","volume":"48","author":[{"given":"Jes\u00fas","family":"Castellanos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julio","family":"Mu\u00f1oz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,4,7]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s002080050159"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s00161-003-0127-3"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"J. M. Ball,\n                      A version of the fundamental theorem for Young measures\n                      , in PDEs and Continuum Models of Phase Transitions, Lecture Notes in Phys. 344, M. Rascle, D. Serre, and M. Slemrod, eds., Springer, New York, 1989, pp. 207\u2013215.","DOI":"10.1007\/BFb0024945"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(2002)128:11(1119)"},{"key":"R5","first-page":"701","volume":"4","author":"Bevan J.","year":"2005","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(99)80169-4"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01557084"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01182325"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01126384"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-003-0238-5"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0312025"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"B. Dacorogna,\n                      Direct Methods in the Calculus of Variations\n                      , Springer-Verlag, New York, 1989.","DOI":"10.1007\/978-3-642-51440-1"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202597000268"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"R. V. Gamkrelidze,\n                      Principles of Optimal Control Theory\n                      , Plenum Press, New York, 1978.","DOI":"10.1007\/978-1-4684-7398-8"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144504446187"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998342829"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(01)00277-2"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(96)00185-X"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"P. Pedregal,\n                      Parametrized Measures and Variational Principles\n                      , Birkh\u00e4user-Verlag, Basel, Switzerland, 1997.","DOI":"10.1007\/978-3-0348-8886-8"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S036301299630080X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(02)00547-4"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(01)00039-7"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-003-0195-z"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"T. Roubicek,\n                      Relaxation in Optimization Theory and Variational Calculus\n                      , W. de Gruyter, Berlin, 1997.","DOI":"10.1515\/9783110811919"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1002\/mma.1670180902"},{"key":"R26","first-page":"427","volume":"7","author":"Roubicek T.","year":"2000","journal-title":"J. Convex Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0944-6532","issn-type":"print"},{"key":"R27","first-page":"447","volume":"8","author":"Stepanov E.","year":"2001","journal-title":"J. Convex Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0944-6532","issn-type":"print"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"L. Tartar,\n                      Nonlocal effects induced by homogenization\n                      , in Partial Differential Equations and the Calculus of Variations, Essays in Honor of E. de Giorgi, Vol. II, Birkh\u00e4user Boston, Cambridge, MA, 1990, pp. 925\u2013938.","DOI":"10.1007\/978-1-4615-9831-2_19"},{"key":"R29","unstructured":"L. C. Young,\n                      Lectures on Calculus of Variations and Optimal Control Theory\n                      , W. B. Saunders, Philadelphia, 1969."}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080740635","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:32:42Z","timestamp":1787319162000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080740635"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":29,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080740635"],"URL":"https:\/\/doi.org\/10.1137\/080740635","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}