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May  2020, 19(5): 2445-2471. doi: 10.3934/cpaa.2020107

## Uniform a priori estimates for elliptic problems with impedance boundary conditions

 1 Inria Sophia Antipolis Méditerranée, 2004 Route des Lucioles, 06902 Valbonne, France 2 Laboratoire J.A. Dieudonné UMR CNRS 7351, Parc Valrose, 06108 Nice Cedex 2, France 3 LAMAV, FR CNRS 2956, Université Polytechnique Hauts-de-France, F-59313 - Valenciennes Cedex 9, France

* Corresponding author

Received  August 2018 Revised  September 2019 Published  March 2020

We derive stability estimates in $H^2$ for elliptic problems with impedance boundary conditions that are uniform with respect to the impedance coefficient. Such estimates are of importance to establish sharp error estimates for finite element discretizations of contact impedance and high-frequency Helm-holtz problems. Though stability in $H^2$ is easily obtained by employing a bootstrap'' argument and well-established result for the corresponding Neumann problem, this strategy leads to a stability constant that increases with the impedance coefficient. Here, we propose alternative proofs to derive sharp and uniform stability constants for domains that are convex or smooth.

Citation: Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
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##### References:
Examples of domains with an obstacle satisfying Assumptions 1 and 2 with $\Gamma_{{\rm{Diss}}}$ convex
Examples of domains with an obstacle satisfying Assumptions 1 and 2 with $\Gamma_{{\rm{Diss}}}$ smooth
Examples of cavities satisfying Assumptions 1 and 2
Exceptional cavity satisfying Theorem 5.1
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