Layer potential method for a Robin problem in Hardy spaces

Abstract

Let Ω be a bounded Lipschitz domain in $R^n$ with $n\ge 3$. Within an appropriate framework, we use the layer potential method to show that the Robin problem$\{\begin{array}{c}-\mathrm{div}\left(A\nabla u\right)=0\text{in }\Omega ,\\ \left(A\nabla u\right)\cdot \nu \left(Q\right)+bu\left(Q\right)=g\in H^p(\partial \Omega ),\\ {\left(\nabla u\right)}^{⁎}\in L^p(\partial \Omega )\end{array}$ is uniquely solvable for all $1-ϵ<p\le 1$, where $ϵ\in (0,\frac{1}{n})$ is a suitable constant depending on the Lipschitz character of ∂Ω and $H^p(\partial \Omega )$ denotes the atomic Hardy space on the boundary of Ω.