Nodal sets of solutions of equations involving magnetic Schroedinger operator in three dimensions

It is well known that the complexity of the nodal set of a function mainly comes from the singular set on which both the function and the gradient vanish. The singular set of a real-valued solution of a linear elliptic equation has been well investigated. For a complex-valued solution of a linear equation involving a magnetic Schroedinger operator, the structure of the nodal set has not been well investigated yet excepted in the two-dimensional case. In this paper we extend the arguments of Garafalo and Lin [Indiana Univ. Math. J. 35, 245-268 (1986)] and of Han [Indiana Univ. Math. J. 43, 983-1002 (1994)] to show that the singular set of such a solution in a three-dimensional domain is countably 1-rectifiable. The functions considered in this paper include the order parameter in the Ginzburg-Landau theory of superconductivity and the eigenfunctions of the magnetic Schroedinger operator