Summary: J. reine angew. Math. 584 (2005), 117--148 Journal fu¨r die reine und
( Walter de Gruyter
Berlin Á New York 2005
Gradient estimates for the pðxÞ-Laplacean system
By Emilio Acerbi and Giuseppe Mingione at Parma
Abstract. We prove Caldero´n and Zygmund type estimates for a class of elliptic
problems whose model is the non-homogeneous pðxÞ-Laplacean system
DuÞ ¼ ÀdivðjFjpðxÞÀ2
Under optimal continuity assumptions on the function pðxÞ > 1 we prove that
loc ) jDujpðxÞ
loc Eq > 1:
Our estimates are motivated by recent developments in non-Newtonian fluidmechanics and
elliptic problems with non-standard growth conditions, and are the natural, ``non-linear''
counterpart of those obtained by Diening and Ru°zicka  in the linear case.