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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.

35P15Estimation of eigenvalues and upper and lower bounds for PD operators
49R50Variational methods for eigenvalues of operators (MSC2000)
51M16Inequalities and extremum problems (geometry)
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