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Gradient estimates for the $$p(x)$$- Laplacian system. (English) Zbl 1093.76003
Summary: We prove Calderón and Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous $$p(x)$$-Laplacian system $-\text{div}\bigl( |Du|^{p(x)-2}Du\bigr)=-\text{div}\bigl( |F|^{p(x)-2}F\bigr).$ Under optimal continuity assumptions on the function $$p(x)>1$$ we prove that $|F|^{p(x)}\in L^q_{\text{loc}} \Rightarrow|Du|^{p(x)}\in L^q_{\text{loc}} \quad\forall q>1.$ Our estimates are motivated by recent developments in non-Newtonian fluid mechanics and elliptic problems with non-standard growth conditions, and are the natural, “nonlinear” counterpart of those obtained by L. Diening and M. Růžička [ibid. 563, 197–220 (2003; Zbl 1072.76071)] in the linear case.

##### MSC:
 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics 35J15 Second-order elliptic equations
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