On the eigenvalues of a Robin problem with a large parameter

Filinovskiy, Alexey. "On the eigenvalues of a Robin problem with a large parameter." Mathematica Bohemica 139.2 (2014): 341-352. <http://eudml.org/doc/261924>.

@article{Filinovskiy2014,
abstract = {We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega $, $\{\partial u\}/\{\partial \nu \}+\alpha u=0$ on $\partial \Omega $ where $\Omega \subset \mathbb \{R\}^n$, $n \ge 2$ is a bounded domain and $\alpha $ is a real parameter. We investigate the behavior of the eigenvalues $\lambda _k (\alpha )$ of this problem as functions of the parameter $\alpha $. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative $\lambda _1^\{\prime \}(\alpha )$. Assuming that the boundary $\partial \Omega $ is of class $C^2$ we obtain estimates to the difference $\lambda _k^D-\lambda _k(\alpha )$ between the $k$-th eigenvalue of the Laplace operator with Dirichlet boundary condition in $\Omega $ and the corresponding Robin eigenvalue for positive values of $\alpha $ for every $k=1,2,\dots $.},
author = {Filinovskiy, Alexey},
journal = {Mathematica Bohemica},
keywords = {Laplace operator; Robin boundary condition; eigenvalue; large parameter; Laplace operator; Robin boundary condition; eigenvalues; large parameter},
language = {eng},
number = {2},
pages = {341-352},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the eigenvalues of a Robin problem with a large parameter},
url = {http://eudml.org/doc/261924},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Filinovskiy, Alexey
TI - On the eigenvalues of a Robin problem with a large parameter
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 341
EP - 352
AB - We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega $, ${\partial u}/{\partial \nu }+\alpha u=0$ on $\partial \Omega $ where $\Omega \subset \mathbb {R}^n$, $n \ge 2$ is a bounded domain and $\alpha $ is a real parameter. We investigate the behavior of the eigenvalues $\lambda _k (\alpha )$ of this problem as functions of the parameter $\alpha $. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative $\lambda _1^{\prime }(\alpha )$. Assuming that the boundary $\partial \Omega $ is of class $C^2$ we obtain estimates to the difference $\lambda _k^D-\lambda _k(\alpha )$ between the $k$-th eigenvalue of the Laplace operator with Dirichlet boundary condition in $\Omega $ and the corresponding Robin eigenvalue for positive values of $\alpha $ for every $k=1,2,\dots $.
LA - eng
KW - Laplace operator; Robin boundary condition; eigenvalue; large parameter; Laplace operator; Robin boundary condition; eigenvalues; large parameter
UR - http://eudml.org/doc/261924
ER -