The Neumann Problem and Helmholtz Decomposition in Convex Domains

Abstract

We show that the Neumann problem for Laplace's equation in a convex domain $Ω$ with boundary data in $L^p(\partialΩ)$ is uniquely solvable for $1<p<\infty$. As a consequence, we obtain the Helmholtz decomposition of vector fields in $L^p(Ω, \mathbb{R}^d)$.

17 pages