On Some Classes of Nonuniformly Elliptic Equations

Abstract

We have called the quasi-linear equation

$${a_{ij}}(x,u,{u_x}){u_{{x_i}{x_j}}} + a(x,u,{u_x}) = 0$$

(1)

uniformly elliptic if

$$\nu (|u|){\xi ^2} \leqslant {a_{ij}}(x,u,p){\xi _i}{\xi _j} \leqslant \mu (|u|){\xi ^2}$$

where ν(τ) and µ(τ) are continuous positive functions for τ≥ 0. Such equations have been, in particu-lar, the fundamental object of investigations concerning their solvability for the case of boundary prob-lems “in the large” and the matter of obtaining various a priori estimates for all their possible solu-tions (see [2]). One of the principal a priori estimates is the estimate $\operatorname{m} \mathop a\limits_\Omega x|{u_x}|$ . The methods we have given for obtaining this estimate (see §§3,4, Chap. 4, and §2, Chap. 6 of [2]) have also been applied, with corresponding modifications, to some classes of nonuniformly elliptic equations. A series of such classes was singled out in papers [3]–[7]. We point out here still another class of such equations.

The results given here appeared in [1] in abbreviated form.