  In this note we will provide a gradient estimate for harmonic maps from a
complete noncompact Riemannian manifold with compact boundary (which we call
"Kasue manifold") into a simply connected complete Riemannian manifold with
non-positive sectional curvature. As a consequence, we can obtain a Liouville
theorem. We will also show the nonexistence of positive solutions to some
linear elliptic equation on Kasue manifold.
