  An extension of the lower-bound lemma of Boggio is given for the weak forms
of certain elliptic operators, which have partially Dirichlet and partially
Neumann boundary conditions, and are in general nonlinear. Its consequences and
those of an adapted Hardy inequality for the location of the bottom of the
spectrum are explored in corollaries wherein a variety of assumptions are
placed on the shape of the Dirichlet and Neumann boundaries.
