Nonlinear gradient estimates for elliptic equations of general type (Q690639)

The interesting paper under review deals with a very general model of nonvariational elliptic equations of \(p\)-Laplacian type \[ \text{div\,}\mathbf{a}(x,Du)=\text{div\,}(|\text\textbf{f}|^{p-2}\text\textbf{f})\quad \text{in}\;\Omega, \] where \(\mathbf{f}\in L^p(\Omega,\mathbb R^n).\) Assuming \(\mathbf{f}\in L^q_{\text{loc}}(\Omega,\mathbb R^n)\) with \(q>p,\) the authors prove \(W^{1,q}_{\text{loc}}\)-estimates for the solutions under the assumption that for each point and for each sufficiently small scale the nonlinearity is suitably close to a vector of the gradient in the BMO space. Extensions of the results to Orlicz spaces framework are also presented. The machinery used relies on Hardy-Littlewood maximal function and variants of the Vitali covering lemma.