Non occurrence of the Lavrentiev gap for a class of nonautonomous functionals (Q6618008)

The paper investigates a central issue in the calculus of variations -- the Lavrentiev phenomenon. This phenomenon describes the situation where the infimum of a functional over Sobolev functions is strictly lower than over Lipschitz functions, creating a ``gap'' between the two spaces. The authors address this problem in the context of multidimensional scalar problems, introducing sufficient conditions under which the gap does not occur. Their findings unify and simplify existing conditions in the literature while broadening the scope of applicability.\N\NA key contribution of the paper is the introduction of a new hypothesis, \(H_{\alpha, L}\), which discards the Lavrentiev gap without relying on traditional constraints like the \(p-q\) growth condition or \(N\)-functions. The hypothesis ensures that the functional behaves consistently across the Sobolev and Lipschitz spaces, even for general classes of Lagrangians that depend on the spatial variable, a Sobolev function, and its gradient. By doing so, the authors significantly extend the range of problems where the Lavrentiev phenomenon can be ruled out.\N\NThe authors provide rigorous theoretical foundations to support their claims, proving results that apply to a wide variety of functionals. They also include numerous classical and novel examples to illustrate the flexibility of their approach. These examples demonstrate the conditions under which the hypothesis \(H_{\alpha, L}\) holds, covering cases like double-phase integrands and integrands with variable exponents. Additionally, the authors' framework enables the study of integrands that do not satisfy standard growth or continuity conditions.\N\NThis work represents a substantial contribution to the field by offering a unifying perspective on a long-standing problem. It not only consolidates known results but also opens avenues for addressing functionals previously considered outside the reach of existing methodologies. While the paper focuses primarily on scalar problems, it leaves room for future research into vectorial settings and more nuanced dependencies of the functionals. Overall, this paper is a valuable resource for researchers exploring regularity and minimization in calculus of variations.