On Variations of Neumann Eigenvalues of p-Laplacian Generated by Measure Preserving Quasiconformal Mappings

In this paper we study variations of the first non-trivial eigenvalues of the two-dimensional $p$-Laplace operator, $p>2$, generated by measure preserving quasiconformal mappings $\phi :\mathbb{D}\to \mathrm{\Omega }$, $\Omega \subset\mathbb R^2$. This study is based on the geometric theory of composition operators on Sobolev spaces with applications to sharp embedding theorems. By using a sharp version of the reverse H"older inequality we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors type domains.