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

Abstract EN

Let Ω ⊂ ℝ n be a nonsmooth convex domain and let f be a distribution in the atomic Hardy space H a t p ( Ω ) ; we study the Schrödinger equations - div ( A ∇ u ) + V u = f in Ω 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 Ω with the Dirichlet boundary condition for n / ( n + 1 ) < p ≤ 2 .

Some fundamental pointwise estimates for the Green function are also given.