In this abortive paper, we will discuss how to find lower bounds for lowest eigenvalues. The aim will be to get across the basic ideas, so we will discuss only the simplest cases, and we won't worry about matters like exactly how many derivatives we are demanding of our functions. The work described here is joint work with Bob Brooks and Bob Kohn. I seem to recall that I was changing the notation around when I turned to other things, so there may be some inconsistencies. Also, you'll note the the story breaks off abruptly near the end.
Continuous version Consider a nice bounded domain in the Euclidean plane.
The lowest eigenvalue is simple, and the corresponding eigenfunction does not change sign in the interior of . Then Thus On the other hand, if we put then and Corollary. Let Then Proof. Let Then Proof. The lowest eigenvalue is simple, and the corresponding eigenfunction does not change sign in the interior of .