Estimation of the height of caved and fractured zones above a longwall panel along with the stability conditions of the goaf area are very crucial to determine the abutment stresses, ground subsidence, and face support as well as designing the surrounding gates and intervening pillars. In this work, the height of caving-fracturing zone above the mined panel is considered as the height of destre...

This note is concerned with the Bianchi–Egnell inequality, which quantifies stability of Sobolev and its generalization to fractional exponents s ∈ ( 0 , d 2 ) $s \in (0, \frac{d}{2})$ . We prove that in dimension ⩾ $d \geqslant 2$ best constant c B E = inf f H ̇ R ∖ M ∥ − Δ / L S ∗ dist $$\begin{equation*} c_{BE}(s) \inf _{f \dot{H}^s(\mathbb {R}^d) \setminus \mathcal {M}} \frac{\Vert (-\Delta ...