Local regularity for anisotropic magnetic operators with general codimension singularities

Abstract

We study local regularity properties of solutions to stationary anisotropic magnetic Schrödinger equations in $\mathbb{R}^d$, $d \ge 2$, arising from singular magnetic potentials concentrated along manifolds of general codimension $2 \le n \le d$. The magnetic interaction is modeled through a covariant gradient of the form \[ \nabla_m u = (iM\nabla + A)u, \] where $M^T M$ is a uniformly elliptic matrix encoding anisotropy and $A$ is a magnetic potential with critical Hardy-type scaling along the $n$-codimensional singular set $Σ_0$; that is, $A\sim \mathrm{dist}(\cdot,Σ_0)^{-1}$. We establish local Hölder $C^{0,α}$ and Schauder $C^{1,α}$ estimates for weak solutions via a blow-up analysis adapted to the magnetic structure. The regularity is deeply influenced by the combined effect of anisotropy and the singular magnetic potential, which determines the spectrum of the limiting spherical Laplace-Beltrami operator arising in the blow-up at the singular set. Our model is motivated by the study of magnetic potentials generated by shrinking solenoids onto an axis $Σ_0$, in the three-dimensional setting $d=3$, $n=2$, leading to Aharonov-Bohm-type (AB) models. In this framework, we show that the geometry of the solenoidal loops plays a crucial role: in particular, any deviation from planar cross-sections orthogonal to $Σ_0$ induces a twofold effect. On the one hand, it breaks the ideal AB configuration, in the sense that the magnetic field outside the solenoid is no longer vanishing. On the other hand, it yields an unexpected regularizing mechanism on the wave functions, through a positive shift in the eigenvalues of the asymptotic spectral problem. This purely three-dimensional effect is consistent with our $C^{1,α}$ regularity.