Comptes Rendus
Control theory
Some Regularity Properties on Bolza problems in the Calculus of Variations
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 205-218.

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.

Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.220
Classification: 49J05, 49J15, 49J30, 49N60
Julien Bernis 1; Piernicola Bettiol 1; Carlo Mariconda 2

1 Université de Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, 6 Avenue Victor Le Gorgeu, 29200 Brest, France
2 University of Padua, Dipartimento di Matematica “Tullio Levi-Civita”, Padova 35121, Italy
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2022__360_G3_205_0,
     author = {Julien Bernis and Piernicola Bettiol and Carlo Mariconda},
     title = {Some {Regularity} {Properties} on {Bolza} problems in the {Calculus} of {Variations}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {205--218},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.220},
     language = {en},
}
TY  - JOUR
AU  - Julien Bernis
AU  - Piernicola Bettiol
AU  - Carlo Mariconda
TI  - Some Regularity Properties on Bolza problems in the Calculus of Variations
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 205
EP  - 218
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.220
LA  - en
ID  - CRMATH_2022__360_G3_205_0
ER  - 
%0 Journal Article
%A Julien Bernis
%A Piernicola Bettiol
%A Carlo Mariconda
%T Some Regularity Properties on Bolza problems in the Calculus of Variations
%J Comptes Rendus. Mathématique
%D 2022
%P 205-218
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.220
%G en
%F CRMATH_2022__360_G3_205_0
Julien Bernis; Piernicola Bettiol; Carlo Mariconda. Some Regularity Properties on Bolza problems in the Calculus of Variations. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 205-218. doi : 10.5802/crmath.220. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.220/

[1] Giovanni Alberti; Francesco Serra Cassano Non-occurrence of gap for one-dimensional autonomous functionals, Calculus of variations, homogenization and continuum mechanics (Marseille, 1993) (Series on Advances in Mathematics for Applied Sciences), Volume 18, World Scientific, 1994, pp. 1-17 | MR | Zbl

[2] Luigi Ambrosio; Oscar Ascenzi; Giuseppe Buttazzo Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl., Volume 142 (1989), pp. 301-316 | DOI | MR | Zbl

[3] John M. Ball; Victor J. Mizel One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation, Arch. Ration. Mech. Anal., Volume 90 (1985), pp. 325-388 | DOI | MR | Zbl

[4] Julien Bernis; Piernicola Bettiol; Carlo Mariconda Lipschitz regularity of the value function and solutions to the Hamilton–Jacobi equation for nonautonomous problems in the calculus of variations (2020) (in preparation)

[5] Piernicola Bettiol; Carlo Mariconda A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers, J. Differ. Equations, Volume 268 (2020) no. 5, pp. 2332-2367 | DOI | MR | Zbl

[6] Piernicola Bettiol; Carlo Mariconda Regularity and necessary conditions for a Bolza optimal control problem, J. Math. Anal. Appl., Volume 489 (2020) no. 1, p. 124123, 17 | DOI | MR | Zbl

[7] Piernicola Bettiol; Carlo Mariconda A Du Bois-Reymond convex inclusion for non-autonomous problems of the Calculus of Variations and regularity of minimizers, Appl. Math. Optim., Volume 83 (2021), pp. 2083-2107 | DOI | Zbl

[8] Piernicola Bettiol; Carlo Mariconda Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem (2021) (submitted)

[9] Giuseppe Buttazzo; Mariano Giaquinta; Stefan Hildebrandt One-dimensional variational problems. An introduction, Oxford Lecture Series in Mathematics and its Applications, 15, Clarendon Press, 1998, viii+262 pages

[10] Arrigo Cellina The classical problem of the calculus of variations in the autonomous case: relaxation and Lipschitzianity of solutions, Trans. Am. Math. Soc., Volume 356 (2004) no. 1, pp. 415-426 | DOI | MR | Zbl

[11] Arrigo Cellina; Alessandro 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, Volume 20 (2003) no. 6, pp. 911-919 | DOI | Numdam | MR | Zbl

[12] Arrigo Cellina; Giulia Treu; Sandro Zagatti On the minimum problem for a class of non-coercive functionals, J. Differ. Equations, Volume 127 (1996) no. 1, pp. 225-262 | DOI | MR | Zbl

[13] Francis H. Clarke An indirect method in the calculus of variations, Trans. Am. Math. Soc., Volume 336 (1993) no. 2, pp. 655-673 | DOI | MR | Zbl

[14] Francis H. Clarke; Richard B. Vinter Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Am. Math. Soc., Volume 289 (1985), pp. 73-98 | DOI | MR | Zbl

[15] Gianni Dal Maso; Hélène Frankowska Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton–Jacobi equations, Appl. Math. Optim., Volume 48 (2003) no. 1, pp. 39-66 | MR | Zbl

[16] Basilio Manià Sopra un esempio di Lavrentieff, Boll. Unione Mat. Ital., Volume 13 (1934), pp. 146-153 | Zbl

[17] Carlo Mariconda Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems (2020) (submitted)

[18] Carlo Mariconda; Giulia Treu Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth, Calc. Var. Partial Differ. Equ., Volume 29 (2007) no. 1, pp. 99-117 | DOI | MR | Zbl

[19] Isidor P. Natanson Theory of functions of real variable, Frederick Ungar Publishing Co., 1955, 277 pages (translated by Leo F. Boron with the collaboration of Edwin Hewitt)

[20] James Serrin; Dale E. Varberg A general chain rule for derivatives and the change of variables formula for the Lebesgue integral, Am. Math. Mon., Volume 76 (1969), pp. 514-520 | DOI | MR | Zbl

Cited by Sources:

Comments - Policy


Articles of potential interest

On Lipschitz regularity of minimizers of a calculus of variations problem with non locally bounded Lagrangians

Marc Quincampoix; Nadia Zlateva

C. R. Math (2006)


Hölder continuity of solutions to a basic problem in the calculus of variations

Pierre Bousquet; Carlo Mariconda; Giulia Treu

C. R. Math (2008)


Continuité lipschitzienne des solutions d'un problème en calcul des variations

Pierre Bousquet; Francis Clarke

C. R. Math (2006)