Second-Order Regularity Estimates for Singular Schrödinger Equations on Convex Domains

Abstract

Let $\mathrm{\Omega }\subset {ℝ}^n$ be a nonsmooth convex domain and let $f$ be a distribution in the atomic Hardy space $H_{at}^p(\mathrm{\Omega })$; we study the Schrödinger equations $-\mathrm{div}(A\nabla u)+Vu=f$ in $\mathrm{\Omega }$ with the singular potential $V$ and the nonsmooth coefficient matrix $A$. We will show the existence of the Green function and establish the $L^p$ integrability of the second-order derivative of the solution to the Schrödinger equation on $\mathrm{\Omega }$ with the Dirichlet boundary condition for . Some fundamental pointwise estimates for the Green function are also given.