PROPERTIES OF ELASTIC SYMBOLS AND CONSTRUCTION OF SOLUTIONS OF THE DIRICHLET PROBLEM

MISHIO KAWASHITA AND HIDEO SOGA^{∗} FACULTY OF EDUCATION, IBARAKI UNIVERSITY MITO IBARAKI, 310-8512 JAPAN

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} ≥ 0}.

Let a_{ij} (i, j, = 1, ⋯, n) be n×n-matrices satisfying appropriate (elastic) assumptions, and consider the equation

\{ \begin{matrix}(σ^{2}I + ∑_{i,j=1}^{n} a_{ij}∂_{x_{i}}∂_{x_{j}})u(x) = 0  in R_{+}^{n}, \\ u|_{x_{n}=0} = e^{iξ′x′}v, \end{matrix}

where σ is an arbitrary real parameter (≠ 0 and fixed). Our purpose is to construct the bounded solution for any ξ′ ∈ R^{n−1} and v ∈ C^{n}.

Taking a root z̃ of the equation det(I − L(η′, z)) = 0 in z and a vecter v ∈ Ker (I −L(η′, z̃)) (where ξ′ = ση′ and L(η) = ∑_{i,j=1}^{n} a_{ij}η_{i}η_{j}), we can make a solution of the form e^{iξ′x′}e^{iσz̃x_{n}} ṽ. But it is not guaranteed that ṽ can be equal to v, or that sum of the solutions of that form over the different roots z̃ can cover any boundary data e^{iξ′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 z̃ of det(I − L(η′, z)) coincides with dim Ker(I − L(η′, z̃)). As another assertion, even if it does not coincide with the dimension, we can construct the solution of a certain form different from e^{iξ′x′}e^{iσz̃x_{n}} ṽ. 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(η′, z)). The matrix (I − L(η′, z))^{−1} is a meromorphic function of z and has poles only at the roots of det(I − L(η′, 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.

∗SPEAKER