On the Lavrentiev phenomenon for autonomous second-order integrands
The so-called Lavrentiev phenomenon is related to the question of whether the infimum of a functional on the class of absolutely continuous functions is strictly lower than the infimum of the same functional on a dense subspace of this class. This phenomenon was first introduced by M. A. Lavrentiev in 1926. The article clarifies the result presented by \textit{F. H. Clarke} and \textit{R. B. Vinter} [Trans. Am. Math. Soc. 291, 73-98 (1985; Zbl 0563.49009)] ensuring the non-existence of a Lavrentiev gap in the case of autonomous first-order integrands on \(\mathbb{R}^ n\)-valued \(u\). The authors present theorems and related examples showing that for certain autonomous second- order integrands a Lavrentiev gap occurs for a dense subclass \(W^{2,p} (0,1)\) of the function space \(W^{2,1}(0,1)\) if one considers nonnegative functions \((u\geq 0)\). On the other hand, the gap typically fails to take place for smooth integrands over the full function space (i.e, if one allows functions possessing fixed sign). By considering additive perturbations of the original problem, the authors obtain a class of fully regular problems exhibiting the Lavrentiev gap.
scientific article; zbMATH DE number 635832
scientific article