The Neumann Problem and Helmholtz Decomposition in Convex Domains

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 $L^p(Ω, \mathbb{R}^d)$.

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