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\begin{document}

\newcommand{\rot}{\mathop{\rm rot}\nolimits}
\newcommand{\diver}{\mathop{\rm div}\nolimits}
%\def\sls{\hbox{\tiny c}\kern-.09em\raise.3ex\hbox{\tiny/}
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\title{INITIAL-BOUNDARY VALUE PROBLEM FOR MAXWELL EQUATION WITH
LEONTOVICH BOUNDARY CONDITION}
\author{M.A.Antonets, L.V.Ponomareva}{
Radiophysical Research Institute, Nizhny Novgorod, Russia}

The initial-boundary value problem for Maxwell equations
  without space charge
\begin{eqnarray*}
{\partial E\over \partial t}=\frac c \varepsilon \rot H,&&
\frac{\partial H}{\partial t}=-\frac c \mu \rot E+j,\\
{\rm div}\, H=0,&& \diver E=0
\end{eqnarray*}
with Leontovich boundary conditions
$\vec H_\tau=Z\vec E_\tau$, where $Z=\displaystyle{\left(
\begin{array}{cc} 0&-\displaystyle{\sqrt{\varepsilon\over\mu}}\\
\displaystyle{\sqrt{\varepsilon\over\mu}}&0
\end{array}
\right)}$, on a surface  $S$, defined by the equation
$x_1=f(x_2,x_3)$ is discussed.

 For the resolvent of corresponding operator an representation, which
is close to M.G.Krein formula, is constructed.
$$ \hat R_z(\lambda)=\hat R(\lambda)-\hat R(\lambda)Q^*C(\lambda)Q
\hat R(\lambda),$$
where $\hat R(\lambda)$ is the resolvent of the Maxwell equations in the
free space, $Q$ is the operator of the evaluation of the
boundary values of the tangent components of vector fields.
$$ Q^*v=\{0,v_1\delta_S,v_2\delta_S\},\qquad v=\{v_1,v_2\}.$$
$$ C(\lambda)=\left(
\begin{array}{cc}
(Zm_1-m_2)^{-1}Z&0\\ 0&(Zm_2+m_1)^{-1}
\end{array} \right),
$$ where $m_1$ =A8 $m_2$ are blocks of operator $m=Q\hat R(\lambda)Q^*=
\displaystyle{ \left(
\begin{array}{cc}
m_1(\lambda)&m_2(\lambda)\\ -m_2(\lambda)&m_1(\lambda)
\end{array}
\right)}$.

 Some estimates for the resolvent norms, which imply the
 existence of unique solution of initial-boundary  value problem are
 obtained.

This work is supported by the  State High Education
Committee of Russian Federation, Grant 95-0-21-77.

\end{document}
