Generalized characteristics and Lax-Oleinik operators: global theory
  For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics. 
Piermarco Cannarsa, Wei Cheng
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