Gradient estimates for anisotropic elliptic equations

Gary M. Lieberman

Abstract

We study a class of elliptic equations with different degeneracies in different directions such as $$\left( |u_x|^{m-2}u_x\right)_x + \left( |u_y|^{q-2}u_y\right)_y =0$$ with unequal parameters $m$ and $q$, both greater than one. We show that any bounded solution of such an equation must have a bounded gradient. The ideas also apply to a much more general class of degenerate equations, which we describe in some detail.

Article information

Source
Adv. Differential Equations Volume 10, Number 7 (2005), 767-812.

Dates
First available in Project Euclid: 18 December 2012