The Neumann Problem and Helmholtz Decomposition in Convex Domains

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