The ${\dot{W}}^{-1,p}$ Neumann problem for higher order elliptic equations

Abstract

We solve the Neumann problem in the half space ${\mathbb{R}}_{+}^{n+1}$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in the negative smoothness space ${\dot{W}}^{-1,p}$, where $\max (0,\frac{1}{2}-\frac{1}{n}-\epsilon )<\frac{1}{p}<\frac{1}{2}$. 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 $L^p$ or ${\dot{W}}^{\pm 1,p}$ 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.

47 pages

References in corpus (1)

• Layer potentials for general linear elliptic systems

Cited by in corpus (1)

• The $L^p$ Neumann problem for higher order elliptic equations