  The Laplace operator acting on antisymmetric tensor fields in a
$D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary
conditions (absolute and relative ones, in the language of Gilkey) are
considered. The eigenfuctions of the operator are found explicitly for all
values of $D$. Using in a row a number of basic techniques, as Mellin
transforms, deformation and shifting of the complex integration contour, and
pole compensation, the zeta function of the operator is obtained. From its
expression, in particular, $\zeta (0)$ and $\zeta'(0)$ are evaluated exactly. A
table is given in the paper for $D=3, 4, ...,8$. The functional determinants
and Casimir energies are obtained for $D=3, 4, ...,6$.
