Methods and Applications of Analysis

Volume 19 (2012)

Number 2

On the existence of smooth solutions for fully nonlinear elliptic equations with measurable “coefficients” without convexity assumptions

Pages: 119 – 146



Nicolai V. Krylov (Department of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.)


We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term one can find an approximating equation which has a unique continuous and having the second derivatives locally bounded solution in a given smooth domain with smooth boundary data. 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 derivatives.


fully nonlinear elliptic equations, Bellman’s equations, finite differences

2010 Mathematics Subject Classification

35J60, 39A14

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