The Neumann Problem and Helmholtz Decomposition in Convex Domains

TL;DR

This paper proves the unique solvability of the Neumann problem for Laplace's equation in convex domains with boundary data in L^p spaces, leading to a Helmholtz decomposition of vector fields in L^p.

Contribution

It establishes the solvability of the Neumann problem in convex domains for a range of p and derives the Helmholtz decomposition in L^p spaces, extending classical results.

Findings

Unique solvability of Neumann problem for 1<p<∞

Helmholtz decomposition in L^p spaces

Extension of classical results to 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 $L^p(\Omega, \mathbb{R}^d)$.