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

We consider the nonlinear singular integral magnetic field equation $R{\bf{M}} = h{\bf{M}} + A{\bf{M}} = {\bf{H}}_a $, in the Hilbert space of vector-functions ${\bf{L}}^2 ( \Omega )$, where ${\bf{M}}$ is the magnetization vector, $( {h{\bf{M}}} )(x) = g [ {{\bf{M}}( x ),x]}$ is the total field, and $({\bf {AM}})(x) = ( - 1/(4\pi )){\operatorname{grad}}\,{\operatorname{div}}\smallint _\Omega ({\bf M}(y)/r)dy$. We prove that: (i) A is bounded, with $\parallel A\parallel = 1$; (ii) A is self-adjoint; (iii) A is positively semidefinite, with $( {A{\bf{M}},{\bf{M}}} )\geqq 0$.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.