Mathematical Study of the Nonlinear Singular Integral Magnetic Field Equation. III
Mark Jay Friedman · SIAM Journal on Mathematical Analysis · 1981
We extend the results of Part I [SIAM J. Appl. Math., 39 (1980), pp. 14–20] on the spectrum of the singular integral operator \[ (A{\bf M})(x) = - \frac{1}{{4\pi }}{\operatorname{grad}}{\operatorname{div}}\int_\Omega {\frac{{{\bf M}(y)}}{r}dy.} \] As an application we obtain an estimate of the lower bound of the spectrum of the magnetic field operator $R{\bf M} = h{\bf M} + A{\bf M}$ from ${\bf L}^2 \Omega $ into the subspace J of generalized solenoidal vector-functions from ${\bf L}^2 $. Here ${\bf M}$ is the magnetization vector, $h{\bf M} = ({{\bf M} / {(\mu (M,x) - 1)}})(M = | {\bf M} |)$ is the total field, $A{\bf M}$ is the induced field, and $\Omega $ is a simply connected domain in $R_3 $.