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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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