# Universal singular sets in the calculus of variations.

*Archive of Rational Mechanics* **190** Number 3 (2008), 371-424.

## Abstract

For regular one-dimensional variational problems, Ball and
Nadirashvilli introduced the notion of the universal singular set of a Lagrangian
\(L\) and established
its topological negligibility. This set
is defined to be the set of all points in the plane through which the graph
of some absolutely continuous \(L\)-minimizer passes with infinite
derivative.

Motivated by Tonelli's partial regularity results, the
question of the size of the universal singular set in measure
naturally arises.
Here we show that universal singular sets are characterized by being
essentially purely unrectifiable --- that is, they intersect most
Lipschitz curves in sets of zero length and that any compact purely unrectifiable
set is contained within the universal singular set of some smooth Lagrangian with
given superlinear growth. This gives examples of universal
singular sets of Hausdorff dimension two, filling the gap between previously known
one-dimensional examples and Sychev's result that universal singular sets are
Lebesgue null.

We show that some smoothness of the Lagrangian is necessary
for the topological size estimate, and investigate the relationship between
growth of the Lagrangian and the existence of (pathological) rectifiable pieces in the
universal singular set.

We also show that Tonelli's partial regularity result is stable in that the energy of
a `near' minimizer \(u\) over the set where it has large derivative is controlled by how far \(u\) is from being a minimizer.

Both [PDF, 371k] and [PS, 658k] versions of the preprint are available.

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**Created:***8 September 2006*

**Modified:***22 January 2009*