Lavrentiev Phenomenon and integral representation of the lower semicontinuous envelope for functionals with non convex and non continuous Lagrangians

In this thesis, we investigate classical integral functionals in the Calculus of Variations and study conditions on the Lagrangian that guarantee the absence of the Lavrentiev Phenomenon. In particular, we focus on Lagrangians that are non-convex, possibly highly discontinuous, and unbounded with respect to the gradient variable. Our first step to prove the non-occurrence of the Lavrentiev Phenomenon is proving the representation of the lower semicontinuous envelope of a functional defined in the space of Lipschitz functions as an integral functional whose Lagrangian is given by the bipolar of the original one. Specifically, we adapt a technique presented in the classical book by Ekeland and Temam for the bounded case, and a refinement of a method due to Cellina for the unbounded case. The integral representation also allows us to apply recent results to the non-convex setting. We establish the integral representation of the lower semicontinuous envelope and the corresponding absence of the Lavrentiev Phenomenon under very weak assumptions in the autonomous case and under suitable anti-jump conditions on the spatial variable in the non-autonomous case. Furthermore, we prove the strong convergence of the approximating sequence for minimizers or under specific growth assumptions on the Lagrangian. These results provide new insights into the interplay between regularity assumptions, relaxation methods, and the avoidance of the Lavrentiev Phenomenon, thereby extending recent advances to a broader class of variational problems.