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

TL;DR

This paper refines Helmholtz decompositions for bounded domains in R^3, showing functions have zero mean locally, and applies these results to improve bounds on Maxwell-type constants using Poincare inequalities.

Contribution

It introduces a local zero mean property for Helmholtz decompositions and improves bounds on Maxwell-type constants via Poincare inequalities in convex domains.

Findings

Functions in Helmholtz spaces have zero mean locally in decompositions.

Derived a Poincare-type inequality with an upper bound for the constant.

Improved upper bounds for Maxwell-type constants using the Payne-Weinberger bound.

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.