Existence and unicity for stationary electromagnetic fields with dissipative boundary conditions
The authors prove existence, uniqueness, and continuous dependence upon the data (the function J) of the a.e. (or weak) solution of the time- harmonic Maxwell equations \(rot E+i\omega \mu H=0,\quad rot H-i\omega \epsilon E=J,\quad x\in \Omega \subset R^ 3,\) with the impedance, or dissipative, boundary condition ''à la Leontovich-Schelkunov-Graffi'' \(E_{\tan g}=\lambda H\wedge n,\) \(x\in \partial \Omega\), where n is the outer normal to \(\partial \Omega\), \(E_{\tan g}\) is the tangential component of E on \(\partial \Omega\), and the impedance \(\lambda\) has positive real part. The case where \(\lambda ={1\over2}(1+i)[2\mu_ 2\omega /\sigma_ 2]^{{1\over2}}\) (with \(\mu_ 2\) and \(\sigma_ 2\) the magnetic permeability and electrical conductivity of the material boundary, respectively) is treated in detail. Well-posedness of the problem fails for \(\lambda =0\) perfectly conducting surface, \(\sigma_ 2=\infty)\).
scientific article; zbMATH DE number 3863751