We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.
Bibliographical noteFunding Information:
The authors are sincerely grateful to the referee for his very quick report and helpful comments on an earlier version of the manuscript. H. Dong was partially supported by NSF Grant DMS-1056737. N. V. Krylov was partially supported by NSF Grant DMS-1160569
- Bellman's equations
- Finite differences
- Fully nonlinear parabolic equations