The singular sets of degenerate and nonlocal elliptic equations on Poincaré-Einstein manifolds

Abstract

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincaré-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincaré-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $\epsilon$-regularity result for degenerate/singular elliptic operators on Poincaré-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincaré-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators.