Composition Operators on Sobolev Spaces and Neumann Eigenvalues

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the $p$-Laplace operator in cusp domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, are given.