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Interior $W^{2,p}$ regularity for the sigma-$2$ equation
We prove that continuous $2$-convex solutions of the $σ_{2}$ equation with a density bounded above and below by positive constants belong to $W^{2,1}_{\mathrm{loc}}$ in every dimension. If, in addition, the density is continuous, the solutions belong to $W^{2,p}_{\mathrm{loc}}$ for every $1<p<\infty
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