Nontangential Maximal Function estimates for the elliptic Mixed Boundary Value Problem with variable coefficients

Abstract

We consider an elliptic operator $L$ with variable, merely bounded, and measurable coefficients on a Lipschitz domain, and study solutions to $Lu=0$ that attain given Neumann and Dirichlet-regularity data on different parts of the boundary. The boundary data lies in $L^p$ or $W^{1,p}$ respectively, and we show nontangential maximal function estimates of the gradient of the solution. This mixed boundary value problem generalizes the pure Dirichlet, regularity, and Neumann problem with rough boundary data in $L^p$, and the already established mixed boundary value problem for the Laplacian.