Mathematical Study of the Nonlinear Singular Integral Magnetic Field Equation. I

We consider the nonlinear singular integral magnetic field equation RM = hM + AM = Hₐ, in the Hilbert space of vector-functions L² ( Ω ), where M is the magnetization vector, ( hM )(x) = g [ M( x ),x] is the total field, and (AM)(x) = ( - 1/(4π ))grad\,div _Ω ( M(y)/r)dy. We prove that: (i) A is bounded, with ∥ A∥ = 1; (ii) A is self-adjoint; (iii) A is positively semidefinite, with ( AM,M ) 0. Uniqueness is proved in case h is strictly monotone; existence of R- 1 and its continuity are proved in case h is strongly monotone, continuous and bounded. In this case the Galerkin method (and, if magnetic-material is also isotropic, the Ritz method) is shown to yield a numerical solution of the equation.