Local regularity for anisotropic magnetic operators with general codimension singularities

TL;DR

This paper investigates the local regularity of solutions to anisotropic magnetic Schrödinger equations with singular potentials along manifolds of various codimensions, revealing how geometry and anisotropy influence solution smoothness.

Contribution

It establishes new local Hölder and Schauder regularity estimates for solutions, accounting for anisotropy and singular magnetic potentials, with applications to Aharonov-Bohm models in three dimensions.

Findings

Regularity depends on anisotropy and magnetic potential singularities.

Deviations from planar solenoids induce spectral shifts and regularization.

The results include Hölder and Schauder estimates for weak solutions.

Abstract

We study local regularity properties of solutions to stationary anisotropic magnetic Schr\"odinger 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 $\Sigma_0$; that is, $A\sim \mathrm{dist}(\cdot,\Sigma_0)^{-1}$. We establish local H\"older $C^{0,\alpha}$ and Schauder $C^{1,\alpha}$ 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 $\Sigma_0$, 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 $\Sigma_0$ 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 $C^{1,\alpha}$ regularity.