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
can be characterized by a variational principle. We are mainly interested in obtaining non-trivial upper bounds for
. 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.