Various classes of non-uniformly elliptic (and parabolic) equations of second order of the form
for all solutions u ( x ) of which m a n Iuzl can be estimated by maxn [uI and m a a R JuxJ, were discussed in [I] (see also [2]).l The method used was introduced in [3]. In the same paper a method was suggested for obtaining local estimates of Iu,I , i.e., estimates of maxn, Iu,I and the distance d(fZ', 82) of 0' c fZ from the boundary 82. In a series of papers (concerning these see [4] and [5]) we have shown that this method is applicable to the whole class of uniformly elliptic and parabolic equations. In the present paper we investigate the possibility of applying it to non-uniformly elliptic and parabolic equations. I t turns out that it is applicable, roughly speaking, to those classes of [I] for which the order of nonuniformity of the quadratic form aij(x, u, is less than two. The first part of this paper is devoted to the proof of this assertion.
In the second part we analyze a different method of obtaining local estimates for 1u. I which is applicable to elliptic equations of the form in terms of m a n IuI and embraces such interesting cases as equations for the mean curvature of a 1 We shall use the notation 677