Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet   conditions in part
  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. 
Evans M. Harrell