The smallest eigenvalue of
certain boundary problems for second order linear elliptic partial differential
equations increases to infinity as the domain in question shrinks to the empty set.
The object of this note is to formulate and prove an analogous result for linear
elliptic differential operators L of general even order. Specifically, let G(t) be a
bounded domain in n-dimensional Euclidean space, and suppose that G(t) has
thickness t (in a sense which will be precisely defined below). Let λ0(t) be the
smallest eigenvalue of a boundary problem associated with L and G(t). It
will be shown that λ0(t) increases to infinity as t tends to zero from the
right.
