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Bais\\'on, Joan Orobitg, Raffaella Giova","submitted_at":"2016-03-17T16:27:07Z","abstract_excerpt":"We study nonlinear elliptic equations in divergence form $${\\operatorname{div}}{\\mathcal A}(x,Du)={\\operatorname{div}}G.$$ When ${\\mathcal A}$ has linear growth in $Du$, and assuming that $x\\mapsto{\\mathcal A}(x,\\xi)$ enjoys $B^\\alpha_{\\frac{n}\\alpha, q}$ smoothness, local well-posedness is found in $B^\\alpha_{p,q}$ for certain values of $p\\in[2,\\frac{n}{\\alpha})$ and $q\\in[1,\\infty]$. In the particular case ${\\mathcal A}(x,\\xi)=A(x)\\xi$, $G=0$ and $A\\in B^\\alpha_{\\frac{n}\\alpha,q}$, $1\\leq q\\leq\\infty$, we obtain $Du\\in B^\\alpha_{p,q}$ for each $p<\\frac{n}\\alpha$. 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