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

Abstract

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We consider the following classical autonomous variational problem minimize F ( v ) = ∫ a b f ( v ( x ) , v ' ( x ) ) x ̣ : v ∈ A C ( [ a , b ] ) , v ( a ) = α , v ( b ) = β , where the Lagrangianf 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/272812>.

@article{Cupini2011,
abstract = {We consider the following classical autonomous variational problem\[ \textrm \{minimize\,\} \left\lbrace F(v)=\int \_a^b f(v(x),v^\{\prime \}(x))\ x̣\,:\,v\in AC([a,b]), \;v(a)=\alpha ,\; v(b)=\beta \right\rbrace ,\]where the Lagrangianf 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; DuBois-Reymond's necessary condition},
language = {eng},
number = {1},
pages = {222-242},
publisher = {EDP-Sciences},
title = {Monotonicity properties of minimizers and relaxation for autonomous variational problems},
url = {http://eudml.org/doc/272812},
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
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 1
SP - 222
EP - 242
AB - We consider the following classical autonomous variational problem\[ \textrm {minimize\,} \left\lbrace F(v)=\int _a^b f(v(x),v^{\prime }(x))\ x̣\,:\,v\in AC([a,b]), \;v(a)=\alpha ,\; v(b)=\beta \right\rbrace ,\]where the Lagrangianf 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; DuBois-Reymond's necessary condition
UR - http://eudml.org/doc/272812
ER -

References

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