An iteration procedure for constructing minimax and viscous solutions to Hamilton-Jacobi equations. (English. Russian original) Zbl 0902.35023

Dokl. Math. 53, No. 3, 416-419 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 348, No. 6, 736-739 (1996).
The main result is Theorem 1 providing a certain iterative-type characterization of the so called minimax solutions in [A. I. Subbotin, “Generalized solutions of first order PDEs” (Birkhäuser, Boston, 1995; Zbl 0820.35003)], which, under certain conditions, coincide with the better known viscosity solutions of problems of the form \[ u_t + H(t,x,u,D_xu)=0, \;(t,x)\in G:=(0,\theta)\times \mathbb{R}^n, \;u(\theta ,x)= \sigma (x). \]
The epigraph \(U\) and, respectively, the hypograph, \(V\) of a minimax solution are shown to be of the forms \[ U= \bigcap_{i\geq 0}U_i, \quad V= \bigcap_{i\geq 0}V_i \] where \(U_i,V_i\) are recurrently defined by an associated differential (“characteristic”) inclusion that is said to generalize the classical system of characteristics.


35F20 Nonlinear first-order PDEs
35A35 Theoretical approximation in context of PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games


Zbl 0820.35003