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On the singularities of the viscosity solutions to Hamilton-Jacobi- Bellman equations. (English) Zbl 0612.70016
Indiana Univ. Math. J. (to appear).
The connection between the local structure of the singularities of solutions to Hamilton-Jacobi-Bellman equation and the set of their superdifferentials is investigated. First, we study the geometry of the singularities of functions in the class Lip(\(\alpha\),\(\Omega)\). This class consists of functions whose second order differences satisfy a one- sided estimate, and for \(\alpha =1\) a function is in Lip(1,\(\Omega)\) if and only if it is the sum of a smooth function and a concave one. Due to a classical one sided second-derivative estimate for the viscosity solutions of Hamilton-Jacobi-Bellman equations, this is a natural class in the study of these equations. As a corollary, we show that the singularities of the solutions to a Cauchy problem propagate in time.

MSC:
70H20 Hamilton-Jacobi equations in mechanics