Geometric characterization of minimax solutions of the Hamilton-Jacobi equation (Q1419553)

Summary: The minimax solution is a weak solution of a Cauchy problem for the Hamilton-Jacobi equation, constructed from a generating family (quadratic at infinity) of its geometric solution. In this paper we give a new construction of the minimax in terms of Morse theory, and we show its stability by small perturbations of the generating family. Then we show that the max-min solution coincides with the minimax solution. Finally, we consider the wave front corresponding to the geometric solution as the graph of a multi-valued solution of the Cauchy problem, and we give a geometric criterion to find the graph of the minimax.