Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture

Abstract

Abstract. In this paper we obtain estimates for the first nontrivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains Ω ⊂ R 2 . This study is based on a quasiconformal version of the universal weighted Poincaré-Sobolev inequalities obtained in our previous papers for conformal weights. The suggested weights in the present paper are Jacobians of quasiconformal mappings. The main technical tool is the theory of composition operators in relation with the Brennan's Conjecture for (quasi)conformal mappings.

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

“…In our recent works [8,9,16,21,22] the spectral stability problem and the lower estimates of Neumann eigenvalues in planar domains were studied. In space domains our results are more modest.…”

Space quasiconformal mappings and Neumann eigenvalues in fractal type domains

“…In the recent works [2,3,15,16,17] the spectral stability problem and lower estimates of Neumann eigenvalues in planar domains were considered. In [18] spectral estimates in space domains using the theory of weak p-quasiconformal mappings were obtained.…”

Space quasiconformal composition operators with applications to Neumann eigenvalues

“…In the case of composition operators generated by quasiconformal mappings the main estimates of norms of embedding operators from Sobolev spaces with first derivatives into Lebesgue spaces were reformulated in the terms of integrals of quasiconformal derivatives [14,15,17]. This type of global integrability of quasiconformal derivatives depends on the quasiconformal geometry of domains [9].…”