Perfectly Matched Layers on Cubic Domains for Pauli’s Equations

This article proves the wellposedness of the boundary value problem that arises when PML algorithms are applied to Pauli’s equations with a three dimensional rectangle as computational domain. The absorptions are positive near the boundary and zero far from the boundary so are always <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> -dependent. At the flat parts of the boundary of the rectangle, the natural absorbing boundary conditions are imposed. The difficulty addressed is the analysis of the resulting variable coefficient problem on the rectangular solid with its edges and corners. The Laplace transform is analysed. We derive an additional boundary condition that is automatically satisfied and yields a coercive Helmholtz boundary value problem on smoothed boundaries with uniform estimates justifying the limit of vanishing smoothing. Uniqueness is reduced by an analyticity argument to our uniqueness theorem for symmetric hyperbolic problems in domains with trihedral corners [16]. This yields the first stability proof with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> -dependent absorptions on a domain whose boundary is not smooth.