# 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 -

## References

top- L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl.142 (1989) 301–316.
- V.I. Bogachev, Measure Theory, Volume I. Springer-Verlag, Berlin, Germany (2007).
- B. Botteron and B. Dacorogna, Existence of solutions for a variational problem associated to models in optimal foraging theory. J. Math. Anal. Appl.147 (1990) 263–276.
- B. Botteron and B. Dacorogna, Existence and nonexistence results for noncoercive variational problems and applications in ecology. J. Differ. Equ.85 (1990) 214–235.
- B. Botteron and P. Marcellini, A general approach to the existence of minimizers of one-dimensional noncoercive integrals of the calculus of variations. Ann. Inst. Henri Poincaré, Anal. non linéaire8 (1991) 197–223.
- P. Celada and S. Perrotta, Existence of minimizers for nonconvex, noncoercive simple integrals. SIAM J. Control Optim.41 (2002) 1118–1140.
- A. Cellina, The classical problem of the calculus of variations in the autonomous case: relaxation and lipschitzianity of solutions. Trans. Amer. Math. Soc.356 (2004) 415–426.
- A. Cellina and A. Ferriero, Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case. Ann. Inst. Henri Poincaré, Anal. non linéaire20 (2003) 911–919.
- A. Cellina, G. Treu and S. Zagatti, On the minimum problem for a class of non-coercive functionals. J. Differ. Equ.127 (1996) 225–262.
- L. Cesari, Optimization: theory and applications. Springer-Verlag, New York, USA (1983).
- F.H. Clarke, An indirect method in the calculus of variations. Trans. Amer. Math. Soc.336 (1993) 655–673.
- G. Cupini, M. Guidorzi and C. Marcelli, Necessary conditions and non-existence results for autonomous nonconvex variational problems. J. Differ. Equ.243 (2007) 329–348.
- B. Dacorogna, Direct methods in the Calculus of Variations, Applied Mathematical Sciences78. Second edition, Springer, Berlin, Germany (2008).
- G. Dal Maso and H. Frankowska, Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton-Jacobi equations. Appl Math Optim.48 (2003) 39–66.
- E. De Giorgi, G. Buttazzo and G. Dal Maso, On the lower semicontinuity of certain integral functionals. Atti Accad. Naz. Lincei, VIII. Ser.74 (1983) 274–282.
- I. Ekeland and R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications1. North Holland, Amsterdam, The Netherlands (1976).
- N. Fusco, P. Marcellini and A. Ornelas, Existence of minimizers for some nonconvex one-dimensional integrals. Port. Math.55 (1998) 167–185.
- O.A. Ladyzhenskaya and N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering46. Academic Press, New York-London (1968).
- C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers. SIAM J. Control Optim.40 (2002) 1473–1490.
- C. Marcelli, Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints. Trans. Amer. Math. Soc.360 (2008) 5201–5227.
- P. Marcellini, Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessità. Rend. Mat. Appl.13 (1980) 271–281.
- A. Ornelas, Existence of scalar minimizers for nonconvex simple integrals of sum type. J. Math. Anal. Appl.221 (1998) 559–573.
- A. Ornelas, Existence and regularity for scalar minimizers of affine nonconvex simple integrals. Nonlinear Anal.53 (2003) 441–451.
- A. Ornelas, Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable. Nonlinear Anal.67 (2007) 2485–2496.
- J.P. Raymond, Existence and uniqueness results for minimization problems with nonconvex functionals. J. Optim. Theory Appl.82 (1994) 571–592.

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