  The squared Laplace operator acting on symmetric rank-two tensor fields is
studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this
fourth-order elliptic operator is obtained provided that such tensor fields and
their first (or second) normal derivatives are set to zero at the boundary.
Strong ellipticity of the resulting boundary-value problems is also proved.
Mixed boundary conditions are eventually studied which involve complementary
projectors and tangential differential operators. In such a case, strong
ellipticity is guaranteed if a pair of matrices are non-degenerate. These
results find application to the analysis of quantum field theories on manifolds
with boundary.
