# Boundary Value Problems for the Magnetic Laplacian in Semiclassical Analysis

## TL;DR

This paper extends boundary value problem analysis for the magnetic Laplacian in semiclassical analysis, establishing uniform estimates in Lipschitz domains, generalizing classical results to magnetic operators.

## Contribution

It provides the first uniform nontangential maximal function estimates for the magnetic Laplacian in Lipschitz domains in the semiclassical regime.

## Key findings

- Established uniform boundary estimates for magnetic Laplacian
- Extended classical Laplacian results to magnetic operators
- Results are valid even for smooth domains

## Abstract

This paper is concerned with the magnetic Laplacian $P^h (\A)=(h D+\A)^2$ in semiclassical analysis, where $h$ is a semiclassical parameter. We study the $L^2$ Neumann and Dirichlet problems for the equation $P^h(\A)u=0$ in a bounded Lipschitz domain $\Omega$. Under the assumption that the magnetic field $\nabla \times \A$ is of finite type on $\overline{\Omega}$, we establish the nontangential maximal function estimates for $(h D+\A)u$, which are uniform for $0< h< h_0$. This extends a well-known result due to D. Jerison and C. Kenig for the Laplacian in Lipschitz domains to the magnetic Laplacian in the semiclassical setting. Our results are new even for smooth domains.