On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains

Abstract

Abstract. In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The "free vibrating membrane" corresponds to the Neumann-Laplace operator in bounded plane domains. In this paper we obtain estimates for the first non-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This case includes convex domains, John domains etc... On the base of our estimates we conjecture that the eigenvalues of the Neumann-Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates which is based on weighted Poincaré-Sobolev inequalities, obtained by the authors recently.

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

“…Consider, for example, the unit square Q ⊂ R 2 . Then Q is a conformal α-regular domain for 2 < α ≤ 4 [16] and is a quasiconformal α-regular domain for all 2 < α ≤ ∞ because the unit square Q is quasiisometrically equivalent to the unit disc D. Remark 1.2. Because ϕ : Ω → D is a quasiconformal mapping, then integrability of the derivative is equivalent to integrability of the Jacobian:…”

Conformal spectral stability estimates for the Neumann Laplacian

“…Recall that for conformal regular domains for any the Sobolev type inequality holds for any function (see ).…”

“…In was proved (see, also, ) that in such domains the embedding operator is compact. Hence in the conformal regular planar domains the spectrum of the Neumann Laplacian is discrete and can be written in the form of a non‐decreasing sequence where each eigenvalue is repeated as many times as its multiplicity.…”

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Key words eigenvalue problem, elliptic equations, conformal mappings, quasidiscs MSC (2010) 47A75, 47B25, 35J40, 35P15We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains ⊂ C. Conformal regular domains support the Poincaré-Sobolev inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.

IntroductionThis paper is devoted to stability estimates for the eigenvalues of the Neumann Laplacianin conformal regular planar domains ⊂ C.In [21] was proved (see, also, [22]) that in such domains the embedding operatoris compact. Hence in the conformal regular planar domains ⊂ C the spectrum of the Neumann Laplacian is discrete and can be written in the form of a non-decreasing sequencewhere each eigenvalue is repeated as many times as its multiplicity.

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“…holds for any function g ∈ W 1,2 ( ) (see [21]). Using this inequality we prove the following weighted Poincaré inequality for the unit disc:…”