Variational Problems with Nonconvex, Noncoercive, Highly Discontinuous Integrands: Characterization and Existence of Minimizers
Abstract
We consider the functional $F(v)=\int_a^b f(t,v^\prime(t)) dt $ in ${\cal H}_p=\{ v\in W^{1,p}: v(a)=0, v(b)=d\}$, $p\in [1,+\infty]$. Under only the assumption that the integrand is ${\cal L}\otimes {\cal B}_n$-measurable, we prove characterizations of strong and weak minimizers both in terms of the minimizers of the relaxed functional and by means of the Euler--Lagrange inclusion.
As an application, we provide necessary and sufficient conditions for the existence of the minimum, expressed in terms of a limitation on the width of the slope d.
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SIAM Journal on Control and Optimization
Pages: 1473 - 1490
ISSN (online): 1095-7138
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Copyright © 2002 Society for Industrial and Applied Mathematics.
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Published online: 26 July 2006
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