We study the structure of optimal solutions for a class of constrained, sec
ond order variational problems on bounded intervals. We show that, for inte
rvals of length greater than some positive constant, the optimal solutions
are bounded in C-1 by a bound independent of the length of the interval. Fu
rthermore, for sufficiently large intervals, the 'mass' and 'energy' of opt
imal solutions are almost uniformly distributed.