On the structure of universal singular sets for Tonelli minimisers

We consider the first variational problem rigorously solved by the direct method, in Tonelli's pioneering work on convex and super linear energies in the one dimensional situation. Beside the existence, he also proved the first partial regularity result for the minimisers. Optimality of his work, i.e. the existence of a large class of singular sets for minimisers was established over a longer period - culminating in the work by John M. Ball and others. There also the notion of universal singular sets (i.e. points singular for some boundary data) was introduced; later studied by M.Sychev. A quite sharp characterisation of these sets will be discussed here.