Boundary-Value Problems for the Squared Laplace Operator

Abstract

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.