# Portugaliae Mathematica

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**Volume 71, Issue 2, 2014, pp. 141–156**

**DOI: 10.4171/PM/1945**

Asymptotic eigenvalue estimates for a Robin problem with a large parameter

Pavel Exner^{[1]}, Alexander Minakov

^{[2]}and Leonid Parnovski

^{[3]}(1) Doppler Institute for Mathematical Physics and Applied Mathematics, Prague, Czech Republic

(2) Mathematical Physics and Applied Mathematics, Prague, Czech Republic

(3) University College London, UK

Robin problem for the Laplacian in a bounded planar domain with a smooth boundary and a large parameter in the boundary condition is considered. We prove a two-sided three-term asymptotic estimate for the negative eigenvalues. Furthermore, improving the upper bound we get a two term asymptotics in terms of the coupling constant and the maximum of the boundary curvature.

*Keywords: *Laplacian, Robin problem, eigenvalue asymptotics

Exner Pavel, Minakov Alexander, Parnovski Leonid: Asymptotic eigenvalue estimates for a Robin problem with a large parameter. *Port. Math.* 71 (2014), 141-156. doi: 10.4171/PM/1945