In some analysis on a domain $\Gamma$, I want to employ a type of inverse estimate $$\|F\|_X \le \frac{k}{\Delta{x}}\|F\|_{L^2(\Gamma)}$$ where $F$ belongs to a finite-dimensional subspace, $\|X\|$ is equivalent to $H^1(\Gamma)$ (i.e. $\|F\|^2_X \le c\|F\|^2_{H^1(\Gamma)}$ for some constant $c$), $k$ is a constant of the inverse estimate, and $\Delta{x}$ is a known value: the size of a computational mesh.
- How can I go about estimating possible values of the constant of inverse inequality $k$?
- Is it possible to ensure the inverse inequality is realistic and useful in practice, without estimating $k$?