REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ

ABSTRACT

In this second paper, we continue our study on the regularity of free boundaries for some fully nonlinear elliptic equations. Our result is if the free boundary is trapped in a sufficiently narrow strip formed by two Lipschitz graphs, then it is also a Lipschitz graph. Combining with the results in Part 1 (see Ref. [Wang]), the free boundary is C ^1,α.