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We construct the Neumann Green function and establish scale invariantregularity estimates for solutions to the Neumann problem for the ellipticoperator $Lu=-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c}\cdot \nabla u+du$ in a Lipschitz domain $\Omega$. We assume that ${\bf A}$ iselliptic and bounded, that the lower order coefficients belong to scaleinvariant Lebesgue spaces, and that either $d\geq{\rm div}\boldsymbol{b}$ in$\Omega$ and $\boldsymbol{b}\cdot\nu\geq 0$ on $\partial\Omega$ in the sense ofdistributions, or the analogous condition for $\boldsymbol{c}$ holds. Wedevelop the $L^2$ theory, construct the Neumann Green function and showestimates in the respective optimal spaces, and show local and global pointwiseestimates for solutions. The main novelty is that our estimates are scaleinvariant, since our constants depend on the lower order coefficients only viatheir norms, and on the Lipschitz domain only via its Lipschitz character.Moreover, our pointwise estimates are shown in the optimal scale invariantsetting for the inhomogeneous terms and the Neumann data.

The Neumann Green function and scale invariant regularity estimates for elliptic equations with Neumann data in Lipschitz domains