Estimates of the nontangential maximal function for solutions of a second-order elliptic equation
This article considers linear second-order elliptic PDEs without lower order terms with \(L^p\)-Dirichlet boundary values on a \(C^1\) boundary. The authors consider weak solutions in \(W^1_{2,\mathrm{loc}}\) where the \(L^p\)-boundary condition is realized in the sense of parallel surfaces. The normal of the boundary and the coefficients in the PDE are assumed to be Dini continuous. They achieve an \(L^p\)-bound for the tangential maximal function on the boundary which they show to be dominated by the \(L^p\)-norm of the boundary data. This result is then used to show that the unique weak solution of the boundary value problem has a nontangential limit at a.e. boundary point. The limit coincides a.e. with the given boundary data.
scientific article; zbMATH DE number 6124984