| Title:
|
On the eigenvalues of a Robin problem with a large parameter (English) |
| Author:
|
Filinovskiy, Alexey |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
341-352 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $. (English) |
| Keyword:
|
Laplace operator |
| Keyword:
|
Robin boundary condition |
| Keyword:
|
eigenvalue |
| Keyword:
|
large parameter |
| MSC:
|
35J05 |
| MSC:
|
35P15 |
| idZBL:
|
Zbl 06362263 |
| idMR:
|
MR3238844 |
| DOI:
|
10.21136/MB.2014.143859 |
| . |
| Date available:
|
2014-07-14T08:38:09Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143859 |
| . |
| Reference:
|
[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. MR 0338535, 10.1007/BF01597075 |
| Reference:
|
[2] Courant, R., Hilbert, D.: Methoden der mathematischen Physik I.German Springer, Berlin (1968). Zbl 0156.23201, MR 0344038 |
| Reference:
|
[3] Daners, D., Kennedy, J. B.: On the asymptotic behaviour of the eigenvalues of a Robin problem.Differ. Integral Equ. 23 (2010), 659-669. Zbl 1240.35370, MR 2654263 |
| Reference:
|
[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. 10.1134/S0012266111110152 |
| Reference:
|
[5] Giorgi, T., Smits, R. G.: Monotonicity results for the principal eigenvalue of the generalized Robin problem.Ill. J. Math. 49 (2005), 1133-1143. Zbl 1089.35038, MR 2210355, 10.1215/ijm/1258138130 |
| Reference:
|
[6] Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators.Birkhäuser, Basel (2006). Zbl 1109.35081, MR 2251558 |
| Reference:
|
[7] Kato, T.: Perturbation Theory for Linear Operators.Springer, Berlin (1995). Zbl 0836.47009, MR 1335452 |
| Reference:
|
[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. Zbl 0656.35012, MR 1133457 |
| Reference:
|
[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. Zbl 0932.35120, MR 1637882, 10.1137/S0036139996308121 |
| Reference:
|
[10] Lou, Y., Zhu, M.: A singularly perturbed linear eigenvalue problem in $C^1$ domains.Pac. J. Math. 214 (2004), 323-334. Zbl 1061.35061, MR 2042936, 10.2140/pjm.2004.214.323 |
| Reference:
|
[11] Mikhaĭlov, V. P.: Partial Differential Equations.Russian Nauka, Moskva (1983). |
| Reference:
|
[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. Zbl 0246.73072, MR 0312800, 10.1007/BF01593087 |
| . |
