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Transactions of the American Mathematical Society

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Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints

Author: Cristina Marcelli
Journal: Trans. Amer. Math. Soc. 360 (2008), 5201-5227
MSC (2000): Primary 49J30, 49J52, 49K30
Published electronically: May 2, 2008
MathSciNet review: 2415071
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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.

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Additional Information

Cristina Marcelli
Affiliation: Department of Mathematical Sciences, Polytechnic University of Marche, Via Brecce Bianche, 60131 Ancona, Italy

Keywords: Constrained variational problems, autonomous Lagrangians, nonsmooth analysis, noncoercive problems, nonconvex problems, necessary and sufficient conditions, DuBois-Reymond condition.
Received by editor(s): November 25, 2004
Received by editor(s) in revised form: June 30, 2006
Published electronically: May 2, 2008
Article copyright: © Copyright 2008 American Mathematical Society