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Article

Keywords:
Laplace operator; Robin boundary condition; eigenvalue; large parameter
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 $.
References:
[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 10.1007/BF01597075 | MR 0338535
[2] Courant, R., Hilbert, D.: Methoden der mathematischen Physik I. German Springer, Berlin (1968). MR 0344038 | Zbl 0156.23201
[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 2654263 | Zbl 1240.35370
[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.
[5] Giorgi, T., Smits, R. G.: Monotonicity results for the principal eigenvalue of the generalized Robin problem. Ill. J. Math. 49 (2005), 1133-1143. MR 2210355 | Zbl 1089.35038
[6] Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser, Basel (2006). MR 2251558 | Zbl 1109.35081
[7] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). MR 1335452 | Zbl 0836.47009
[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 1133457 | Zbl 0656.35012
[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 10.1137/S0036139996308121 | MR 1637882 | Zbl 0932.35120
[10] Lou, Y., Zhu, M.: A singularly perturbed linear eigenvalue problem in $C^1$ domains. Pac. J. Math. 214 (2004), 323-334. DOI 10.2140/pjm.2004.214.323 | MR 2042936 | Zbl 1061.35061
[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 10.1007/BF01593087 | MR 0312800 | Zbl 0246.73072
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