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\title{ Properties of Elastic Symbols and Construction of \\
Solutions of the Dirichlet Problem}
\author{
Mishio KAWASHITA and Hideo SOGA${}^*$\footnote{${}^*$Speaker} \\
Faculty of Education, Ibaraki University \\ 
Mito Ibaraki, 310-8512 Japan
}

\begin{document}
\maketitle

Plane waves are solutions of the redueced wave equation, and play an
important role on various problems. Our talk deals with construction such
waves of the elastic equation in the half-space $\R^n_+ = \{ x=(x',x_n);
x_n\geq 0\}$.

Let $a_{ij}\, (i,j,= 1,\cdots , n)$ be $n$$\times$$n$-matrices satisfying
appropriate (elastic) assumptions, and consider the equation $$
\left\{
\begin{array}{cl}
&(\sigma^2I+\sum^n_{i,j=1}a_{ij}\partial_{x_i}\partial_{x_j})u(x) = 0 
\qquad \mbox{in} \quad \R^n_+, \\
&u\vert_{x_n=0} = e^{i\xi'x'}v,
\end{array}
\right.
$$
where $\sigma$ is an arbitrary real parameter ($\not=$ 0 and fixed). Our purpose
is to construct the bounded solution for any $\xi'\in\R^{n-1}$ and $v\in\C^n$.

Taking a root $\tilde{z}$ of the equation $\mbox{det}(I-L(\eta',z)) =0$ in $z$
and a vecter $v \in \mbox{Ker}$ $(I-L(\eta',\tilde{z}))$ (where
$\xi'=\sigma \eta'$ and $L(\eta ) = \sum^n_{i,j=1}a_{ij}\eta_i\eta_j$), we
can make a solution of the form
$e^{i\xi'x'}e^{i\sigma\tilde{z}x_n}\tilde{v}$. But it is not guaranteed
that $\tilde{v}$ can be equal to $v$, or that sum of the solutions of that
form over the different roots $\tilde{z}$ can cover any boundary data
$e^{i\xi'x'}v$, which we call the completeness. In general systems, the
completeness is not necessarily obtained, and is set as an assumption.

One of our assertions is that we can obtain the completeness for elastic
equations under some assumptions if multiplicity of every non-real root
$\tilde{z}$ of det($I-L(\eta',z)$) coincides with dim Ker($I-L(\eta',
\tilde{z})$). As another assertion, even if it does not coincide with the
dimension,
we can construct the solution of a certain form different from
$e^{i\xi'x'}e^{i\sigma\tilde{z}x_n}\tilde{v}$. These constructions can be
extended to more general situations (general domains, general data, etc.),
and give a basic tool on inverse scattering problems. 

The proofs are based on the complex analysis of the symbol
$(I-L(\eta',z))$. The matrix $(I-L(\eta',z))^{-1}$ is a meromorphic
function
of $z$ and has poles only at the roots of $\mbox{det}(I-L(\eta',z))=0$ at
most. The first assertion
is verified from the fact that every pole becomes simple. The construction
in the second assertion is accomplished by means of the complex integral.
This is similar to that for the Poisson operator of elliptic equations.

\end{document}
