A Short Note on a Weighted Friedrichs Inequality

TL;DR

This paper derives a generalized upper bound for a weighted Friedrichs inequality constant on bounded domains, improving existing bounds for Maxwell constants and demonstrating practical use in error estimation for elliptic problems.

Contribution

It introduces a new upper bound for a weighted Friedrichs constant that extends known bounds and enhances Maxwell constant estimates for convex domains.

Findings

Derived a generalized upper bound for weighted Friedrichs constants.

Improved bounds for Maxwell type constants in convex domains.

Applied the results to a posteriori error estimation in elliptic problems.

Abstract

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.