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.
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@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/
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