A Short Note on Helmholtz Decompositions for Bounded Domains in $\mathbb{R}^3$

Abstract

In this short note we consider several widely used L^2-orthogonal Helmholtz decompositions for bounded domains in R^3. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine this global property into a local equivalent: we show that functions from these spaces have zero mean in every part of specific decompositions of the domain. An application of the zero mean properties is presented for convex domains. We introduce a specialized Poincare-type inequality, and estimate the related unknown constant from above. The upper bound is derived using the upper bound for the Poincare constant proven by Payne and Weinberger. This is then used to obtain a small improvement of upper bounds of two Maxwell-type constants originally proven by Pauly. Although the two dimensional case is not considered, all derived results can be repeated in R^2 by similar calculations.