An eigenvalue problem is considered where the eigenvalue appears in the domain and on the boundary. This eigenvalue problem has a spectrum of discrete positive and negative eigenvalues. The smallest positive and the largest negative eigenvalue $\lambda_{\pm 1}$ can be characterized by a variational principle. We are mainly interested in obtaining non trivial upper bounds for $\lambda_{-1}$. We prove some domain monotonicity for certain special shapes using a kind of maximum principle derived by C. Bandle, J.v. Bellow and W. Reichel in [J. Eur. Math. Soc., 10 (2007), 73-104]. We then apply these bounds to the trace inequality.
References
C. Bandle, J.v. Below and W. Reichel, Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up, Rend. Lincei Mat. Appl., 17 (2006), 35–67. MR2237743C. Bandle, J.v. Below and W. Reichel, Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up, Rend. Lincei Mat. Appl., 17 (2006), 35–67. MR2237743
C. Bandle, J.v. Below and W. Reichel, Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions, J. Eur. Math. Soc., 10 (2007), 73–104. MR2349897 05255323C. Bandle, J.v. Below and W. Reichel, Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions, J. Eur. Math. Soc., 10 (2007), 73–104. MR2349897 05255323
C. Bandle, A Rayleigh-Faber-Krahn inequality and some monotonicity properties for eigenvalue problems with mixed boundary conditions, to appear in Proceedings of the Conference on Inequalities and Applications '07. MR2758991 10.1007/978-3-7643-8773-0_1C. Bandle, A Rayleigh-Faber-Krahn inequality and some monotonicity properties for eigenvalue problems with mixed boundary conditions, to appear in Proceedings of the Conference on Inequalities and Applications '07. MR2758991 10.1007/978-3-7643-8773-0_1
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