Singular Sets and the Lavrentiev Phenomenon

摘要

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