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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints
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by Cristina Marcelli PDF
Trans. Amer. Math. Soc. 360 (2008), 5201-5227 Request permission

Abstract:

We consider the classical autonomous constrained variational problem of minimization of $\int _a^bf(v(t),v’(t)) dt$ in the class $\Omega :=\{v \in W^{1,1}(a,b):$ $v(a)=\alpha , v(b)= \beta , v’(t)\ge 0 \mbox {a.e. in } (a,b) \}$, where $f:[\alpha , \beta ]\times [0,+\infty ) \to \mathbb {R}$ is a lower semicontinuous, nonnegative integrand, which can be nonsmooth, nonconvex and noncoercive.

We prove a necessary and sufficient condition for the optimality of a trajectory $v_0\in \Omega$ in the form of a DuBois-Reymond inclusion involving the subdifferential of Convex Analysis. Moreover, we also provide a relaxation result and necessary and sufficient conditions for the existence of the minimum expressed in terms of an upper limitation for the assigned mean slope $\xi _0=(\beta -\alpha )/(b-a)$. Applications to various noncoercive variational problems are also included.

References
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Additional Information
  • Cristina Marcelli
  • Affiliation: Department of Mathematical Sciences, Polytechnic University of Marche, Via Brecce Bianche, 60131 Ancona, Italy
  • Email: marcelli@dipmat.univpm.it
  • Received by editor(s): November 25, 2004
  • Received by editor(s) in revised form: June 30, 2006
  • Published electronically: May 2, 2008
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 5201-5227
  • MSC (2000): Primary 49J30, 49J52, 49K30
  • DOI: https://doi.org/10.1090/S0002-9947-08-04514-5
  • MathSciNet review: 2415071