Geodesic
On the eigenvalues of a Robin problem with a large parameter
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 341-352

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We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega $, ${\partial u}/{\partial \nu }+\alpha u=0$ on $\partial \Omega $ where $\Omega \subset \mathbb R^n$, $n \geq 2$ 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 $\lambda _1'(\alpha )$. 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 $.
We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega $, ${\partial u}/{\partial \nu }+\alpha u=0$ on $\partial \Omega $ where $\Omega \subset \mathbb R^n$, $n \geq 2$ 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 $\lambda _1'(\alpha )$. 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 $.
DOI : 10.21136/MB.2014.143859
Classification : 35J05, 35P15
Keywords: Laplace operator; Robin boundary condition; eigenvalue; large parameter 
Filinovskiy, Alexey. On the eigenvalues of a Robin problem with a large parameter. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 341-352. doi: 10.21136/MB.2014.143859
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[1] Bandle, C., Sperb, R. P.: Application of Rellich's perturbation theory to a classical boundary and eigenvalue problem. Z. Angew. Math. Phys. 24 (1973), 709-720. | DOI | MR

[2] Courant, R., Hilbert, D.: Methoden der mathematischen Physik I. German Springer, Berlin (1968). | MR | Zbl

[3] Daners, D., Kennedy, J. B.: On the asymptotic behaviour of the eigenvalues of a Robin problem. Differ. Integral Equ. 23 (2010), 659-669. | MR | Zbl

[4] Filinovskiy, A. V.: Asymptotic behavior of the first eigenvalue of the Robin problem. On the seminar on qualitative theory of differential equations at Moscow State University, Differ. Equ. 47 (2011), 1680-1696. DOI:10.1134/S0012266111110152. | DOI

[5] Giorgi, T., Smits, R. G.: Monotonicity results for the principal eigenvalue of the generalized Robin problem. Ill. J. Math. 49 (2005), 1133-1143. | DOI | MR | Zbl

[6] Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser, Basel (2006). | MR | Zbl

[7] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). | MR | Zbl

[8] Kondrat'ev, V. A., Landis, E. M.: Qualitative theory of second order linear partial differential equations. Russian Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 32 (1988), 99-215. | MR | Zbl

[9] Lacey, A. A., Ockendon, J. R., Sabina, J.: Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math. 58 (1998), 1622-1647. | DOI | MR | Zbl

[10] Lou, Y., Zhu, M.: A singularly perturbed linear eigenvalue problem in $C^1$ domains. Pac. J. Math. 214 (2004), 323-334. | DOI | MR | Zbl

[11] Mikhaĭlov, V. P.: Partial Differential Equations. Russian Nauka, Moskva (1983).

[12] Sperb, R. P.: Untere und obere Schranken für den tiefsten Eigenwert der elastisch gestützten Membran. German Z. Angew. Math. Phys. 23 (1972), 231-244. | DOI | MR | Zbl

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