Singular set estimates for solutions to elliptic equations in higher co-dimension
Max Engelstein, Cole Jeznach, Yannick Sire
TL;DR
This work proves sharp boundary singular-set estimates for solutions to a class of degenerate-elliptic equations in domains with higher-codimension boundaries. By flattening the boundary and analyzing a flat-model operator, the authors prove an almost-monotone Almgren-type frequency, classify Λ-homogeneous solutions, and establish a cone-based quantitative stratification that controls the boundary singular set via Minkowski and Hausdorff estimates. A novel cone-aggregation covering argument replaces the traditional cone-splitting, enabling precise size estimates for S(u) near ∂Ω and, through a change of variables, extends to general boundary geometries with sufficient regularity. The results generalize boundary unique continuation in a higher-codimension setting and reveal intricate boundary-behavior differences from the classical codimension-one theory, including the possible inclusion of the boundary in the singular set and non-integer homogeneities.
Abstract
Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero set of solutions. Such estimates also hold at the boundary, provided that the domain is sufficiently regular. In this work, we investigate the boundary behavior of solutions to a class of elliptic equations in the higher co-dimension setting, whose coefficients are neither uniformly elliptic, nor uniformly Lipschitz. Despite these challenges, we are still able to show analogous estimates on the singular set of such solutions near the boundary. Our main technical advance is a variant of the Cheeger-Naber-Valtorta quantitative stratification scheme using cones instead of planes.