V.A. Kozlov, V.G. Mazya, J. Roßmann
Spectral properties of operator pencils generated by elliptic boundary value problems for the Lame system
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 51, 5-24(1997)
35B40 Asymptotic behavior of solutions
35J25 Boundary value problems for second-order, elliptic equations
35J55 Boundary value problems for elliptic systems
Abstract: The present paper is concerned with the spectral properties of operator
pencils generated by boundary value problems for the Lam\'{e} system
\begin{equation} \label{0.1}
\Delta U + \frac{1}{\gamma}\, \mbox{grad\,div}\, U = F
in a cone ${\cal K}.$ Here $U=(U_1,U_2,U_3)$ denotes the displacement
vector and $\gamma$ is a positive constant which is related to the
Poisson ratio $\nu$ via the equality $\gamma=1-2\nu.$
The study of the spectrum of these operator pencils is of great importance
for the description of the behaviour of the solutions near conical points.
It is well-known (see e.g. \cite{kon-67}, \cite{maz/plam-78/1},
\cite{naz/plam-94}) that the solutions of elliptic boundary value
problems in a neighbourhood of a conical point $x^{(0)}$ asymptotically
behave like a linear combination of terms of the form
Notes: Abstract contains the first few lines of text of the paper.