Local gradient estimates for solutions of a simplest regularization of a class of nonuniformly elliptic equations

Abstract

An estimate of$\mathop {\max }\limits_{\Omega '} |u_x^\varepsilon |$, Ω′⊃⊃Ω, for solutions u^ε of the family of equations

$$ - \frac{d}{{dx_i }}\frac{{u_{x_i } }}{{\sqrt {1 + u_x^2 } }} - \varepsilon \Delta u + a(x,u,u_x ) = 0, x \in \Omega ,\varepsilon \in (0,1],$$

with a nondifferentiable lower term a is given. The majorant in the estimate depends on$\mathop {\max }\limits_\Omega |u^\varepsilon |$ and the distance between Ω′ and ∂Ω, and does not depend on ε. This publication is related to [2, 3]. Bibliography: 4 titles.