Geodesic
On conformal spectral gap estimates of the Dirichlet-Laplacian
Algebra i analiz, Tome 31 (2019) no. 2, pp. 189-203.

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We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains $ \Omega \subset \mathbb{R}^2$. With the help of these estimates, we obtain asymptotically sharp inequalities of ratios of eigenvalues in the framework of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains in terms of conformal (hyperbolic) geometry.
Keywords: elliptic equations, Sobolev spaces, conformal mappings. 
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V. Gol'dshtein; V. Pchelintsev; A. Ukhlov. On conformal spectral gap estimates of the Dirichlet-Laplacian. Algebra i analiz, Tome 31 (2019) no. 2, pp. 189-203. http://geodesic.mathdoc.fr/item/AA_2019_31_2_a7/

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