Monotonicity properties of minimizers and relaxation for autonomous variational problems

• Volume: 17, Issue: 1, page 222-242
• ISSN: 1292-8119

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Abstract

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We consider the following classical autonomous variational problem$\mathrm{minimize}\phantom{\rule{0.166667em}{0ex}}\left\{F\left(v\right)={\int }_{a}^{b}f\left(v\left(x\right),{v}^{\text{'}}\left(x\right)\right)\phantom{\rule{4pt}{0ex}}x̣\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}v\in AC\left(\left[a,b\right]\right),\phantom{\rule{0.277778em}{0ex}}v\left(a\right)=\alpha ,\phantom{\rule{0.277778em}{0ex}}v\left(b\right)=\beta \right\},$ 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 existence or non-existence criteria.

How to cite

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Cupini, Giovanni, and Marcelli, Cristina. "Monotonicity properties of minimizers and relaxation for autonomous variational problems." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 222-242. <http://eudml.org/doc/197345>.

@article{Cupini2011,
abstract = { We consider the following classical autonomous variational problem$\textrm\{minimize\,\} \left\\{F(v)=\int\_a^b f(v(x),v'(x))\ \d x\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta \right\\},$ 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. },
author = {Cupini, Giovanni, Marcelli, Cristina},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonconvex variational problems; autonomous variational problems; existence of minimizers; DuBois-Reymond necessary condition; relaxation; nonconvex variational problems; autonomous variational problems; DuBois-Reymond's necessary condition},
language = {eng},
month = {2},
number = {1},
pages = {222-242},
publisher = {EDP Sciences},
title = {Monotonicity properties of minimizers and relaxation for autonomous variational problems},
url = {http://eudml.org/doc/197345},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Cupini, Giovanni
AU - Marcelli, Cristina
TI - Monotonicity properties of minimizers and relaxation for autonomous variational problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 222
EP - 242
AB - We consider the following classical autonomous variational problem$\textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ \d x\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta \right\},$ 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.
LA - eng
KW - Nonconvex variational problems; autonomous variational problems; existence of minimizers; DuBois-Reymond necessary condition; relaxation; nonconvex variational problems; autonomous variational problems; DuBois-Reymond's necessary condition
UR - http://eudml.org/doc/197345
ER -

References

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