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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 C. Bandle, J. von Bellow and 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 Estimates of eigenvalues in context of PDEs
49R50 Variational methods for eigenvalues of operators (MSC2000)
51M16 Inequalities and extremum problems in real or complex geometry

Citations:

Zbl 1167.35012
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