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

Abstract

We construct the Neumann Green function and establish scale invariant regularity estimates for solutions to the Neumann problem for the elliptic operator $Lu=-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$ in a Lipschitz domain $\Omega$. We assume that ${\bf A}$ is elliptic and bounded, that the lower order coefficients belong to scale invariant 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 of distributions, or the analogous condition for $\boldsymbol{c}$ holds. We develop the $L^2$ theory, construct the Neumann Green function and show estimates in the respective optimal spaces, and show local and global pointwise estimates for solutions. The main novelty is that our estimates are scale invariant, since our constants depend on the lower order coefficients only via their norms, and on the Lipschitz domain only via its Lipschitz character. Moreover, our pointwise estimates are shown in the optimal scale invariant setting for the inhomogeneous terms and the Neumann data.