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.

中文翻译：

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高阶椭圆方程的 Ẇ−1,p Neumann 问题

摘要

我们在半空间解决诺依曼问题 ${\mathbb{电阻}}_{+}^{n+1},$对于具有可变自伴随t独立系数和负平滑空间中的边界数据的高阶椭圆微分方程${\dot{宽}}^{-1,磷},$ 在哪里 $\mathrm{最大限度}(0,\frac{1}{2}-\frac{1}{n}-\epsilon )<\frac{1}{磷}<\frac{1}{2}.$我们的论点受到 Shen 论点的启发，并建立在p  = 2的情况下已知的适定性结果上。我们使用相同的技术来建立非切线和平方函数估计，其中输入为L ^p或${\dot{宽}}^{\pm 1,磷}$对于类似的p范围，基于p接近 2 的已知界限；在这种情况下，我们可以放宽自伴随的要求。