On the Tonelli's partial regularity. (Q364235)

The author studies regularity properties of minima of variational problems with one independent variable, i.e. NEWLINE\[NEWLINE \min \left \{\int_a^bL(t,u(t),u'(t)\,dt\: \, u\in W^{1,1}_0(a,b)\right \}. NEWLINE\]NEWLINE The main result is that minima are regular in the sense of Tonelli, meaning that the derivative of \(u\) is continuous as an extended real-valued function outside an exceptional set of measure zero. The novelty is in assuming that \(L\) is invariant under a certain group of \(C^1\) transformations. This allows relaxing the assumption about the dependency of \(L\) on \(u\) to just continuity; a typical assumption needed for this type of result is Lipschitz continuity.