The Ẇ−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.

Keywords:

• Elliptic equation
• good-λ inequality
• higher-order differential equation
• layer potentials
• Neumann problem

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 35J30
• Secondary: 35J40
• 31B10
• 35C15

Acknowledgments

The author would like to thank Steve Hofmann and Svitlana Mayboroda for many useful conversations on topics related to this paper. The author would also like to thank the Mathematical Sciences Research Institute for hosting a Program on Harmonic Analysis, the Instituto de Ciencias Matemáticas for hosting a Research Term on “Real Harmonic Analysis and Its Applications to Partial Differential Equations and Geometric Measure Theory,” and the IAS/Park City Mathematics Institute for hosting a Summer Session with a research topic of Harmonic Analysis, at which many of the results and techniques of this paper were discussed.

Notes

1 There is a minor error in [20, Section 6], namely a forgotten complex conjugate.