Preprint 2001-029

Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Elliptic Equations

Abstract: Using the maximum principle for semicontinuous functions \cite{CrIs:MaxPrinc,CrIsLi:UserGuide}, we prove a general ``continuous dependence on the nonlinearities'' estimate for bounded H\"{o}lder continuous viscosity solutions of fully nonlinear degenerate elliptic equations. Furthermore, we provide existence, uniqueness, and H\"{o}lder 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.

Paper:

Available as PostScript (159 Kbytes) or gzipped PostScript (50 Kbytes; uncompress using gunzip).

Author(s):

Espen R. Jakobsen, <erj@math.ntnu.no>

Kenneth Hvistendahl Karlsen, <kennethk@mi.uib.no>

Publishing information:

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

Comments:

Some correction has been made in the published version.

Submitted by:

<kennethk@mi.uib.no> July 17 2001.