# Monotonicity properties of minimizers and relaxation for autonomous variational problems

Giovanni Cupini; Cristina Marcelli

ESAIM: Control, Optimisation and Calculus of Variations (2011)

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

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topCupini, 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 -

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