Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints

Abstract Consider a positive functional F ⁢ (y): = ∫ t T L ⁢ (s, y ⁢ (s), y ′ ⁢ (s) ) ⁢ 𝑑 s F (y): =ₓ^{TL (s, y (s), y^ (s) ) \, ds}, defined on the space of Sobolev functions W 1, p ⁢ (t, T ; ℝ n) W^{1, p (t, T;R^n) }, where p ≥ 1 p 1. This functional is minimized among functions y that may satisfy one or both endpoint conditions. The Lagrangian L is allowed to assume the value + ∞ +. In numerous applications minimizers may not be explicit or even may not exist. In such circumstances, it is crucial to know that the infimum of F can be approximated using a sequence of Lipschitz functions that meet the given boundary conditions. However, there are instances where this approximation is not feasible, even with polynomial Lagrangians that meet Tonelli’s existence conditions: this situation is referred to as the Lavrentiev phenomenon. Some results are present in the literature if one requires the Lipschitz approximations to preserve just one endpoint constraint or when the Lagrangian is finite valued. The present paper deals with problems with two endpoint constraints and Lagrangians that are allowed to take extended values. As a byproduct, our Lipschitz approximations may preserve given state constraints. The extended-valued case is challenging since the phenomenon may occur even when the Lagrangian is constant on its effective domain, whose topology becomes relevant. Once assumed that the Lagrangian is radial convex on the rays of the last variable, our findings offer new insights, even when the Lagrangian L is real-valued, autonomous, and there are no state constraints.