Stable domains for higher order elliptic operators

TL;DR

This paper proves that domains satisfying a certain capacity condition are automatically stable for higher order elliptic operators across all dimensions, extending previous results with a simpler proof.

Contribution

It establishes that capacity conditions imply stability for higher order elliptic operators in all dimensions, broadening the class of domains covered.

Findings

Capacity condition implies stability for all orders and dimensions.

Includes domains with cusp points, not just smooth ones.

Extends previous results with a simpler proof.

Abstract

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 $m\geqslant 1$ and $p\geqslant 1$, and for any dimension $N\geqslant 1$. In particular, this includes regular enough domains such as $\mathscr{C}^1-$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.