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

TL;DR

This paper develops a constructive method to approximate Sobolev functions with Lipschitz functions, avoiding the Lavrentiev phenomenon under state constraints, and extends previous results to broader classes of Lagrangians and boundary conditions.

Contribution

It provides a new constructive recipe for approximating functions in Sobolev spaces with Lipschitz functions under state constraints, avoiding the Lavrentiev gap, and extends existing results to more general Lagrangians and boundary conditions.

Findings

Constructive approximation sequence converges in energy to the original functional.

Results apply to both autonomous and non-autonomous Lagrangians.

Enlarged class of Lagrangians and boundary conditions for avoiding Lavrentiev phenomenon.

Abstract

Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional (the "energy"), unnecessarily autonomous, defined on the space of Sobolev functions $W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$). We consider the problem of minimizing $F$ among the functions $y$ that possibly satisfy one, or both, end point conditions. In many applications, where the lack of regularity or convexity or growth conditions does not ensure the existence of a minimizer of $F$, it is important to be able to approximate the value of the infimum of $F$ via a sequence of Lipschitz functions satisfying the given boundary conditions. Sometimes, even with some polynomial, coercive and convex Lagrangians in the velocity variable, thus ensuring the existence of a minimizer in the given Sobolev space, this is not achievable: this fact is know as the Lavrentiev phenomenon. The paper deals on the avoidance of the Lavrentiev phenomenon under the validity of a further given state constraint of the form $y(s)\in\mathcal S\subset\mathbb R^n$ for all $s\in [t,T]$. Given $y\in W^{1,p}([t,T];\mathbb R^n)$ with $F(y)<+\infty$ we give a constructive recipe for building a sequence $(y_h)_h$ of Lipschitz reparametrizations of $y$, sharing with $y$ the same boundary condition(s), that converge in energy to $F(y)$. With respect to previous literature on the subject, we distinguish the case of (just) one end point condition from that of both, enlarge the class of Lagrangians that satisfy the sufficient conditions and show that $(y_h)_h$ converge also in $W^{1,p}$ to $y$. Moreover, the results apply also to extended valued Lagrangians whose effective domain is bounded. The results gives new clues even when the Lagrangian is autonomous, i.e., of the form $L(s,y,y')=\Lambda (y,y')$. The paper follows two recent papers \[23, 24] of the author on the subject.