Singular Sets and the Lavrentiev Phenomenon

Abstract

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subset of the line $E$ and an arbitrary superlinearity, there exists a smooth, strictly convex Lagrangian with this superlinear growth, such that all minimizers of the associated variational problem have singular set exactly $E$, but still admit approximation in energy by smooth functions.

18 pages, to appear in Proc. Roy. Soc. Edinburgh Sect A

References in corpus (1)

• A one-dimensional variational problem with continuous Lagrangian and singular minimizer