Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

Abstract

We solve the Neumann problem in the half space R n+1 + , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness spaceẆ −1,p , where max(0, 1 2 − 1 n −ε) < 1 p < 1 2 . Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2.We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in L p orẆ ±1,p for a similar range of p, based on known bounds for p near 2; in this case we may relax the requirement of self-adjointess.