On the asymptotic behavior of solutions of linear second-order boundary-value problems on a semi-infinite strip

This paper treats the asymptotic behavior of solutions of a linear secondorder elliptic partial differential equation defined on a two-dimensional semiinfinite strip. The equation has divergence form and variable coefficients. Such equations arise in the theory of steady-state heat conduction for inhomogeneous anisotropic materials, as well as in the theory of anti-plane shear deformations for a linearized inhomogeneous anisotropic elastic solid. Solutions of such equations that vanish on the long sides of the strip are shown to satisfy a theorem of Phragmén-Lindelöf type, providing estimates for the rate of growth or decay which are optimal for the case of constant coefficients. The results are illustrated by several examples. The estimates obtained in this paper can be used to assess the influence of inhomogeneity and anisotropy on the decay of end effects arising in connection with Saint-Venant's principle.