Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:463

Article

Gol'dshtein, V¹  Pchelintsev, V^1,2,3  Ukhlov, A.¹ 

[1] Ben Gurion Univ Negev, Dept Math, POB 653, IL-8410501 Beer Sheva, Israel

[2] Tomsk Polytech Univ, Dept Math & Informat, Lenin Ave 30, Tomsk 634050, Russia

[3] Tomsk State Univ, Dept Gen Math, Lenin Ave 36, Tomsk 634050, Russia

关键词: Sobolev spaces;    Conformal mappings;    Quasiconformal mappings;    Elliptic equations;   

DOI  :  10.1016/j.jmaa.2018.02.063

来源: Elsevier

【 摘 要 】

In this paper we study integral estimates of derivatives of conformal mappings phi : D -> Omega of the unit disc D subset of C onto bounded domains Omega that satisfy the Ahlfors condition. These integral estimates lead to estimates of constants in Sobolev-Poincare inequalities, and by the Rayleigh quotient we obtain spectral estimates of the Neumann-Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the (quasi)conformal geometry of the domains. Specifically, the lower estimates of the first non-trivial eigenvalues of the Neumann-Laplace operator in some fractal type domains (snowflakes) were obtained. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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