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# 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

DOI: http://dx.doi.org/10.4310/MAA.2012.v19.n2.a2

#### Author

#### Abstract

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.

#### Keywords

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

#### 2010 Mathematics Subject Classification

35J60, 39A14