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

We extend the results of Part I [SIAM J. Appl. Math., 39 (1980), pp. 14–20] on the spectrum of the singular integral operator \[ (A{ M})(x) = - {1}{{4π }}{grad}{div}∫_Ω {{{{ M}(y)}}{r}dy.} \] As an application we obtain an estimate of the lower bound of the spectrum of the magnetic field operator R M = h M + A M from L² Ω into the subspace J of generalized solenoidal vector-functions from L². Here M is the magnetization vector, h M = ( M / (μ (M,x) - 1))(M = | M |) is the total field, A M is the induced field, and Ω is a simply connected domain in R₃.