Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 263, pp. 1-5.

Title: A note on p(x)-harmonic maps

Authors: Bei Wang (Jiangsu Institute of Education, Nanjing, China)
         Yuze Cai (Shazhou Professional Institute of Technology, Jiangsu, China)

Abstract:
 This article is concerned with L^{p(x)} estimates of the gradient of
 p(x)-harmonic maps. It is known that p(x)-harmonic maps are the weak
 solutions of a system with natural growth conditions, but
 it is difficult to use the classical elliptic techniques to find
 gradient estimates. In this article, we use the monotone inequality
 to show that the minimum p(x)-energy can be expressed by the
 L^{p(x)} norm of a gradient of a function Phi, which is a weak
 solution of a single equation.

Submitted April 10, 2013. Published November 29, 2013.
Math Subject Classifications: 35J56, 35J70, 49J20, 58G18.
Key Words: Gradient estimate; p(x)-harmonic map; drill holes; 
           minimum p(x)-energy