Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part. (English) Zbl 1091.35046

Summary: 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.


35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
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