On nonexistence of minimizing solutions of variational problems (Q1075589)

Considered is the minimum problem for a functional of the type \(\int^{T}_{0}L(\gamma (t),{\dot \gamma}(t))dt.\) Here, L is supposed to be a smooth function of its arguments. For the admissible curves \(\gamma\) : [0,T]\(\to {\mathbb{R}}^ n\) the endpoints \(\gamma (0)=x_ 0\) and \(\gamma (T)=x_ 1\) are prescribed. The author is interested in problems without solution where the Lagrangian L does not necessarily satisfy a convexity assumption so that the associated Hamiltonian vector field is no longer continuous everywhere. A class of such problems is studied for arbitrary dimension n. It is shown that the nonexistence of solutions is caused by the presence of some local type of discontinuity in the Hamiltonian vector field.