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

Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture In this paper we obtain lower estimates for the first non-trivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains $$\varOmega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . This study is based on a quasiconformal version of the universal two-weight Poincaré–Sobolev inequalities obtained in our previous papers for conformal weights and its non weighted version for so-called K-quasiconformal $$\alpha$$ α -regular domains. The main technical tool is the geometric theory of composition operators in relation with the Brennan’s conjecture for (quasi)conformal mappings. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bollettino dell'Unione Matematica Italiana Springer Journals

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

Bollettino dell'Unione Matematica Italiana, Volume 11 (2) – May 17, 2017
20 pages

/lp/springer_journal/spectral-estimates-of-the-p-laplace-neumann-operator-and-brennan-s-IOym7USbjo
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1972-6724
eISSN
2198-2759
DOI
10.1007/s40574-017-0127-z
Publisher site
See Article on Publisher Site

### Abstract

In this paper we obtain lower estimates for the first non-trivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains $$\varOmega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . This study is based on a quasiconformal version of the universal two-weight Poincaré–Sobolev inequalities obtained in our previous papers for conformal weights and its non weighted version for so-called K-quasiconformal $$\alpha$$ α -regular domains. The main technical tool is the geometric theory of composition operators in relation with the Brennan’s conjecture for (quasi)conformal mappings.

### Journal

Bollettino dell'Unione Matematica ItalianaSpringer Journals

Published: May 17, 2017

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