A useful subdifferential in the Calculus of Variations

Consider the basic problem in the Calculus of Variations of minimizing an energy functional depending on absolutely continuous functions Under suitable assumptions on the Lagrangian, a well-known result establishes that the minimizers satisfy the Du Bois-Reymond equation. Recent work (cf. Bettiol and Mariconda, 2020 [1], 2023; Mariconda, 2023 [2], 2021, 2024) highlights not only that a Du Bois-Reymond condition for minimizers can be broadened to cover the case of nonsmooth extended valued Lagrangians, but also that a particular subdifferential (associated with the generalized Du Bois-Reymond condition) plays an important role in the approximation of the energy via its values along Lispchitz functions, no matter minimizers exist. A crucial point is establishing boundedness properties of this subdifferential, based on weak local boundedness properties of the Lagrangian. This is the main objective of this paper. Our approach is based on a refined analysis of the metric that can be employed to evaluate the distance from the complementary of the effective domain of the reference Lagrangian. As an application of our findings we show how it is possible to deduce the non-occurrence of the Lavrentiev phenomenon, providing a new general result.