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 \(\varOmega \subset {\mathbb {R}}^2\). 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 \(\alpha \)-regular domains. The main technical tool is the geometric theory of composition operators in relation with the Brennan’s conjecture for (quasi)conformal mappings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Astala, K., Koskela, P.: Quasiconformal mappings and global integrability of the derivative. J. Anal. Math. 57, 203–220 (1991)
Astala, K.: Area distortion of quasiconformal mappings. Acta Math. 173, 37–60 (1994)
Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2008)
Bertilsson, D.: On Brennan’s conjecture in conformal mapping. Doctoral Thesis, Royal Institute of Technology, Stockholm, Sweden (1999)
Brandolini, B., Chiacchio, F., Trombetti, C.: Sharp estimates for eigenfunctions of a Neumann problem. Commun. Partial Differ. Equ. 34, 1317–1337 (2009)
Brandolini, B., Chiacchio, F., Trombetti, C.: Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems. Proc. R. Soc. Edinb. 145A, 31–45 (2015)
Brennan, J.: The integrability of the derivative in conformal mapping. J. Lond. Math. Soc. 18, 261–272 (1978)
Burenkov, V.I., Gol’dshtein, V., Ukhlov, A.: Conformal spectral stability for the Dirichlet–Laplace operator. Math. Nachr. 288, 1822–1833 (2015)
Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20, 253–264 (2013)
Ferone, V., Nitsch, C., Trombetti, C.: A remark on optimal weighted Poincaré inequalities for convex domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23, 467–475 (2012)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)
Gol’dshtein, V.M.: The degree of summability of generalized derivatives of quasiconformal homeomorphisms. Sib. Math. J. 22, 821–836 (1981)
Gol’dshtein, V., Gurov, L.: Applications of change of variables operators for exact embedding theorems. Integral Equ. Oper. Theory 19, 1–24 (1994)
Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361, 3829–3850 (2009)
Gol’dshtein, V., Ukhlov, A.: Brennan’s conjecture for composition operators on Sobolev spaces. Eurasian Math. J. 3(4), 35–43 (2012)
Gol’dshtein, V., Ukhlov, A.: Conformal weights and Sobolev embeddings. J. Math. Sci. (N.Y.) 193, 202–210 (2013)
Gol’dshtein, V., Ukhlov, A.: Brennan’s conjecture and universal Sobolev inequalities. Bull. Sci. Math. 138, 253–269 (2014)
Gol’dshtein, V., Ukhlov, A.: On the first eigenvalues of free vibrating membranes in conformal regular domains. Arch. Ration. Mech. Anal. 221, 893–915 (2016)
Gol’dshtein, V., Ukhlov, A.: Spectral estimates of the \(p\)-Laplace Neumann operator in conformal regular domains. Trans. A. Razmadze Math. Inst. 170(1), 137–148 (2016)
Hedelman, H., Shimorin, S.: Weighted Bergman spaces and the integral spectrum of conformal mappings. Duke Math. J. 127, 341–393 (2005)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford University Press, Oxford (1993)
Hurri-Syrjänen, R., Staples, S.G.: A quasiconformal analogue of Brennan’s conjecture. Complex Var. Theory Appl. 35, 27–32 (1998)
Koskela, P., Onninen, J., Tyson, J.T.: Quasihyperbolic boundary conditions and capacity: Poincaré domains. Math. Ann. 323, 811–830 (2002)
Koskela, P., Reitich, F.: Hölder continuity of Sobolev functions and quasiconformal mappings. Math. Z. 213, 457–472 (1993)
Maz’ya, V.: Sobolev Spaces: With Applications to Elliptic Partial Differential Equations. Springer, Berlin (2010)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Pólya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. 11, 419–433 (1961)
Ukhlov, A.: On mappings, which induce embeddings of Sobolev spaces. Sib. Math. J. 34, 185–192 (1993)
Valtorta, D.: Sharp estimate on the first eigenvalue of the p-Laplacian. Nonlinear Anal. 75, 4974–4994 (2012)
Vodop’yanov, S.K., Gol’dstein, V.M.: Lattice isomorphisms of the spaces \(W^1_n\) and quasiconformal mappings. Sib. Math. J. 16, 224–246 (1975)
Vodop’yanov, S.K., Gol’dshtein, V.M., Reshetnyak, Yu.G.: On geometric properties of functions with generalized first derivatives. Uspekhi Mat. Nauk 34, 17–65 (1979)
Vodop’yanov, S.K., Ukhlov, A.D.: Sobolev spaces and \((P, Q)\)-quasiconformal mappings of Carnot groups. Sib. Math. J. 39, 665–682 (1998)
Vodop’yanov, S.K., Ukhlov, A.D.: Superposition operators in Sobolev spaces. Russ. Math. (Izvestiya VUZ) 46, 11–33 (2002)
Vodop’yanov, S.K., Ukhlov, A.D.: Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I. Sib. Adv. Math. 14, 78–125 (2004)
Zubelevich, J.: An elliptic equation with perturbed \(p\)-Laplace operator. J. Math. Anal. Appl. 328, 1188–1195 (2007)
Acknowledgements
V. Gol’dshtein is supported by the United States-Israel Binational Science Foundation (BSF Grant No. 2014055).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gol’dshtein, V., Pchelintsev, V. & Ukhlov, A. Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture. Boll Unione Mat Ital 11, 245–264 (2018). https://doi.org/10.1007/s40574-017-0127-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40574-017-0127-z