A Short Note on a Weighted Friedrichs Inequality

In this short note we derive, for bounded domains, an upper bound for a Friedrichs type constant in a weighted Friedrichs type inequality. This upper bound generalizes a well known upper bound of the Friedrichs constant. This upper bound is also used to improve an upper bound of a Maxwell type constant for convex domains in $\mathbb{R}^3$. A simple numerical application is also given: we apply the main result in a posteriori error estimation for an elliptic problem.