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Monotonicity properties of minimizers and relaxationfor autonomous variational problems
Published online by Cambridge University Press: 24 March 2010
Abstract
We consider the following classical autonomous variational problem
\[ \textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ \dx\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta\right\},\]
where the Lagrangian f is possibly neither continuous, nor convex, nor coercive.We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existenceor non-existence criteria.
Keywords
- Type
- Research Article
- Information
- ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 222 - 242
- Copyright
- © EDP Sciences, SMAI, 2010
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