Stable domains for higher order elliptic operators

This paper is devoted to prove that any domain satisfying a $({\delta }_0,r_0)-$capacity condition of first order is automatically $(m,p)-$stable for all $m⩾1$ and $p⩾1$, and for any dimension $N⩾1$. In particular, this includes regular enough domains such as ${\mathcal{C}}^1-$domains, Lipchitz domains, Reifenberg flat domains, but is weak enough to also includes cusp points. Our result extends some of the results of Hayouni and Pierre valid only for $N=2,3$, and extends also the results of Bucur and Zolesio for higher order operators, with a different and simpler proof.