The Neumann Problem and Helmholtz Decomposition in Convex Domains

Abstract

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 $L^p(\Omega ,{\mathbb{R}}^d)$.

17 pages

References in corpus (1)

• Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain