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

Abstract

This paper is devoted to investigating the geometric properties of solutions to certain degenerate equations, and their nonlocal counterparts, in the context of Poincaré–Einstein manifolds. The operators under consideration arise in the theory of conformal invariants on the visual boundary of Poincaré–Einstein manifolds. Within this framework, we develop a quantitative differentiation theory that includes a quantitative stratification of the singular set and Minkowski-type estimates for the (quantitatively) stratified singular sets of various solutions of PDEs on both the complete manifold and its conformal infinity. 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 demonstrate a delicate synergy between the geometry of Poincaré–Einstein manifolds and the theory of the corresponding degenerate elliptic operators.