Monotonicity properties of minimizers and relaxation for autonomous variational problems

Cupini, Giovanni; Marcelli, Cristina

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 222-242.

We consider the following classical autonomous variational problem

$$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$

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.

[1] 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. | MR | Zbl

[2] V.I. Bogachev, Measure Theory, Volume I. Springer-Verlag, Berlin, Germany (2007). | MR | Zbl

[3] 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. | MR | Zbl

[4] B. Botteron and B. Dacorogna, Existence and nonexistence results for noncoercive variational problems and applications in ecology. J. Differ. Equ. 85 (1990) 214-235. | MR | Zbl

[5] 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 linéaire 8 (1991) 197-223. | Numdam | MR | Zbl

[6] P. Celada and S. Perrotta, Existence of minimizers for nonconvex, noncoercive simple integrals. SIAM J. Control Optim. 41 (2002) 1118-1140. | MR | Zbl

[7] 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. | MR | Zbl

[8] 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 linéaire 20 (2003) 911-919. | Numdam | MR | Zbl

[9] 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. | MR | Zbl

[10] L. Cesari, Optimization: theory and applications. Springer-Verlag, New York, USA (1983). | MR | Zbl

[11] F.H. Clarke, An indirect method in the calculus of variations. Trans. Amer. Math. Soc. 336 (1993) 655-673. | MR | Zbl

[12] 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. | MR | Zbl

[13] B. Dacorogna, Direct methods in the Calculus of Variations, Applied Mathematical Sciences 78. Second edition, Springer, Berlin, Germany (2008). | MR | Zbl

[14] 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. | MR | Zbl

[15] 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. | EuDML | MR | Zbl

[16] I. Ekeland and R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications 1. North Holland, Amsterdam, The Netherlands (1976). | MR | Zbl

[17] N. Fusco, P. Marcellini and A. Ornelas, Existence of minimizers for some nonconvex one-dimensional integrals. Port. Math. 55 (1998) 167-185. | EuDML | MR | Zbl

[18] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering 46. Academic Press, New York-London (1968). | MR | Zbl

[19] C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers. SIAM J. Control Optim. 40 (2002) 1473-1490. | MR | Zbl

[20] C. Marcelli, Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints. Trans. Amer. Math. Soc. 360 (2008) 5201-5227. | MR | Zbl

[21] 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. | Zbl

[22] A. Ornelas, Existence of scalar minimizers for nonconvex simple integrals of sum type. J. Math. Anal. Appl. 221 (1998) 559-573. | MR | Zbl

[23] A. Ornelas, Existence and regularity for scalar minimizers of affine nonconvex simple integrals. Nonlinear Anal. 53 (2003) 441-451. | MR | Zbl

[24] A. Ornelas, Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable. Nonlinear Anal. 67 (2007) 2485-2496. | MR | Zbl

[25] J.P. Raymond, Existence and uniqueness results for minimization problems with nonconvex functionals. J. Optim. Theory Appl. 82 (1994) 571-592. | MR | Zbl