# On Conformal Spectral Gap Estimates of the Dirichlet-Laplacian Academic Article

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### abstract

• We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains \$\Omega\subset\mathbb R^2\$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-P\'olya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.

### publication date

• January 1, 2018