An estimate for the region of regularity of solutions of certain nonlinear differential inequalities

In the diskx ²+y ²R ² of thex, y-plane we consider the differential inequalityz _xxz_yy–z _xy ² –(1+z _x ^/2 +z _y ^/2 )^k, where the constants >0 andk>1. In the case =1 andk=2 this inequality means that the surfacez(x, y) has Gaussian curvatureK1. Efimov has shown that in this case the radius of the disk has an upper bound. In the present article we establish an analogous upper bound for the radiusR of the disk in which the functionz(x, y) satisfies the differential inequality above.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 19–21.