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Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations. (English) Zbl 1010.35050
The authors are mainly interested in bounded continuous viscosity solutions of fully nonlinear degenerate elliptic equations of the form \[ F\bigl( x,u(x),Du(x), D^2u(x)\bigr)=0 \text{ in }\mathbb{R}^N. \tag{1} \] Under some suitable assumptions on \(F\), they obtain existence, uniqueness, and Hölder continuity results. They also deal with the problem of finding an upper bound on the difference between a viscosity subsolution \(u\) of (1) and a viscosity supersolution \(\overline u\) of \[ \overline F\bigl(u,\overline u(x), D\overline u(x),D^2 \overline u(x)\bigr)= 0\text{ in }\mathbb{R}^N, \] where \(\overline F\) is another nonlinearity satisfying suitable assumptions. Moreover, they present examples of equations which are covered by their results. In particular, an explicit continuous dependence estimate is stated for the second-order Hamilton-Jacobi-Bellman-Isaacs equations associated with zero-sum, two-player stochastic differential games.

MSC:
35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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