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An eigenvalue problem with mixed boundary conditions and trace theorems. (English) Zbl 1141.35040
Summary: 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 {\it C. Bandle}, {\it J. von Bellow} and {\it W. Reichel} in [J. Eur. Math. Soc. (JEMS) 10, No. 1, 73--104 (2008; Zbl 1167.35012)]. We then apply these bounds to the trace inequality.

##### MSC:
 35P15 Estimation of eigenvalues and upper and lower bounds for PD operators 49R50 Variational methods for eigenvalues of operators (MSC2000) 51M16 Inequalities and extremum problems (geometry)
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