An existence result in nonlinear theory of electromagnetic fields (Q1192792)

The material at each place \(x\) in a bounded region \(R\) of the three- dimensional Euclidean space is specified by the following constitutive equations: \[ \zeta=\hat\zeta(E,H),\qquad D=-\partial\hat\zeta/\partial R, \qquad B=-\partial\hat\zeta/\partial H,\tag{1} \] where \(\zeta\) is the enthalpy density, \(E=(E_ 1,E_ 2,E_ 3)\) is the electric intensity, \(H=(H_ 1,H_ 2,H_ 3)\) is the magnetic intensity, \(D=(D_ 1,D_ 2,D_ 3)\) is the electric induction, \(B=(B_ 1,B_ 2,B_ 3)\) is the magnetic induction. From the Maxwell equations it follows \(E_ i=\varphi_{,i}\), \(H_ i=\psi_{,i}\), where \(\varphi\) is the potential of the electric field and \(\psi\) is the magnetic potential. The equations (1) can be written in the form \[ \zeta=\hat\zeta(\varphi_{,i},\psi_{,i}), \qquad D_ i=- \partial\hat\zeta/\partial\varphi_{,i}, \qquad B_ i=- \partial\zeta/\partial\psi_{,i}.\tag{2} \] The boundary-value problem considered consists of finding the functions \(\varphi\) and \(\psi\) which satisfy the equations (2) and \(\text{div}D=\rho\), \(\text{div} B=0\), in \(R\), where \(\rho\) is the density of charge, and the boundary conditions \(\varphi=\bar\varphi\), \(\psi=\bar\psi\) in \(\partial R\), and \(\bar\varphi,\bar\psi\) are prescribed functions. The aim of this paper is to establish an existence theorem for this boundary-value problem by using results on the theory of nonlinear operators.