Geometric lower bounds for the spectrum of elliptic PDEs

An extension of the lower-bound lemma of Boggio is given for the weak forms of certain elliptic operators, which are in general nonlinear and have partially Dirichlet and partially Neumann boundary conditions. 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. * [email protected], School of Mathematics,Georgia Tech, Atlanta, GA 303320160, USA. c ©2004 by the author. Reproduction of this article, in its entirety, by any means is permitted for non–commercial purposes. This work was supported by NSF grant DMS-0204059.