Electron. J. Diff. Eqns., Vol. 2002(2002), No. 39, pp. 1-10.

Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations

Espen R. Jakobsen & Kenneth H. Karlsen

Using the maximum principle for semicontinuous functions [3,4], we prove a general ``continuous dependence on the nonlinearities'' estimate for bounded Holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations. Furthermore, we provide existence, uniqueness, and Holder continuity results for bounded viscosity solutions of such equations. Our results are general enough to encompass Hamilton-Jacobi-Bellman-Isaacs's equations of zero-sum, two-player stochastic differential games. An immediate consequence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations.

Submitted August 8, 2001. Published May 6, 2002.
Math Subject Classifications: 35J60, 35J70, 49L25.
Key Words: fully nonlinear degenerate elliptic equation, viscosity solution, Hamilton-Jacobi-Bellman-Isaacs equation, continuous dependence estimate, vanishing viscosity method, convergence rate

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Espen R. Jakobsen
Department of Mathematical Sciences
Norwegian University of Science and Technology
N-7491 Trondheim, Norway
e-mail: erj@math.ntnu.no
Kenneth H. Karlsen
Department of Mathematics
University of Bergen
Johs. Brunsgt. 12
N-5008 Bergen, Norway
e-mail: kennethk@mi.uib.no

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