Calculus of Variations - Lavrentiev Phenomenon

Lavrentiev Phenomenon

Hilbert was the first to give good conditions for the Euler Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler Lagrange equations in the interior.

However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. For instance the following:

Here a zig zag path gives a better solution than any smooth path and increasing the number of sections improves the solution.