Advances in Differential Equations

Gradient estimates for anisotropic elliptic equations

Gary M. Lieberman

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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

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

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions 35J70: Degenerate elliptic equations


Lieberman, Gary M. Gradient estimates for anisotropic elliptic equations. Adv. Differential Equations 10 (2005), no. 7, 767--812.

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