| Issue |
ESAIM: COCV
Volume 17, Number 1, January-March 2011
|
|
|---|---|---|
| Page(s) | 222 - 242 | |
| DOI | https://doi.org/10.1051/cocv/2010001 | |
| Published online | 24 March 2010 | |
Monotonicity properties of minimizers and relaxation for autonomous variational problems
1
Dipartimento di Matematica,
Università di Bologna,
Piazza di Porta S.Donato 5, 40126 Bologna, Italy. giovanni.cupini@unibo.it
2
Dipartimento di Scienze Matematiche,
Università Politecnica delle Marche, Via Brecce Bianche,
60131 Ancona, Italy. marcelli@dipmat.univpm.it
Received:
20
November
2008
We consider the following classical autonomous variational problem
![\[ \textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ \d x\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta \right\},\]](/articles/cocv/abs/2011/01/cocv0898/cocv0898_tex_eq1.png)
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 existence or non-existence criteria.
Mathematics Subject Classification: 49K05 / 49J05
Key words: Nonconvex variational problems / autonomous variational problems / existence of minimizers / DuBois-Reymond necessary condition / relaxation
© EDP Sciences, SMAI, 2010
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