The Neumann Problem and Helmholtz Decomposition in Convex Domains

By Jun Geng, Zhongwei Shen. 2010

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

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dacefdc387b04ab017f2925722097aba1712e562