The Dirichlet Problem for Nonuniformly Elliptic Equations

and repeated indices indicate summation from 1 to n. The functions a'(x, u, p), a(x, u, p) are defined in QX£ n + 1 . If furthermore for any ikf>0, the ratio of the maximum to minimum eigenvalues of [a(Xy u, p)] is bounded in ÛX( — M, M)XE, Qu is called uniformly elliptic. A solution of the Dirichlet problem Qu = Q, u—(x) on 30. When Qu is elliptic, but not necessarily uniformly elliptic, it is referred to as nonuniformly elliptic. In this case it is well known from two dimensional considerations, that in addition to smoothness of the boundary data 30, (x) and growth restrictions on the coefficients of Qu, geometric conditions on <90 may play a role in the solvability of the Dirichlet problem. A striking example of this in higher dimensions is the recent work of Jenkins and Serrin [4] on the minimal surface equation, mentioned below. The Dirichlet problem for general classes of nonuniformly elliptic equations has been considered by Gilbarg [ l ] , Stampacchia [7], Hartman and Stampacchia [2], Hartman [3], and Motteler [ó]. We announce below some theorems which extend the results of these authors. The detailed proofs will appear elsewhere. The author gratefully acknowledges the encouragement and assistance of Professor David Gilbarg in this work.