Title: Boundary Value Problems for the Magnetic Laplacian in Semiclassical Analysis Show Chinese title

Title: 半经典分析中磁 Laplacian 的边界值问题

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. Show Chinese abstract

Abstract: 本文关注半经典分析中的磁 Laplacian $P^h (\A)=(h D+\A)^2$，其中 $h$是一个半经典参数。 我们研究方程 $P^h(\A)u=0$在有界 Lipschitz 域 $\Omega$中的 $L^2$Neumann 和 Dirichlet 问题。 在假设磁场$\nabla \times \A$在$\overline{\Omega}$上为有限类型的情况下，我们建立了$(h D+\A)u$的非切向极大函数估计，这些估计对于$0< h< h_0$是一致的。 这将 D. Jerison 和 C. Kenig 对 Lipschitz 域中 Laplacian 的一个著名结果扩展到了半经典情况下磁 Laplacian 的情形。 即使对于光滑域，我们的结果也是新的。