Hausdorff measure of critical sets of solutions to magnetic schrödinger equations

Abstract

In this paper, we study the critical set of a complex-valued solution to a Schrodinger equation involving the magnetic field and with a nonlinear term, where the critical set is \({\{x\in\Omega:~\psi(x)=0, ~\nabla\psi(x)=0\}}\) . We consider this equation in a bounded domain of \({\mathbb{R}^3}\) with the boundary condition: \({\nabla _{\mathbf{A}}\psi\cdot \nu=0}\) , and we establish a global 1-dimensional Hausdorff measure estimate for the critical sets. From the proof of global estimates, we find that our methods work as well for more general equations with a magnetic potential.