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\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
Banach J. Math. Anal. 2 (2008), no. 2, 68--75\\
$\frac{\rule{4.55in}{0.05in}}{{}}$\\
{\footnotesize
\textcolor[rgb]{0.65,0.00,0.95}{\textsc{\textbf{\large{B}}anach
\textbf{\large{J}}ournal of \textbf{\large{M}}athematical
\textbf{\large{A}}nalysis}}\\
ISSN: 1735-8787 (electronic)\\
\textcolor[rgb]{0.00,0.00,0.84}{\textbf{http://www.math-analysis.org }}\\
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\title[Eigenvalue estimates]{An eigenvalue problem with mixed boundary conditions
and trace theorems}

\author[C.Bandle]{Catherine Bandle$^1$}

\address{$^{1}$ Department of Mathematics, University of Basel, Rheinsprung 21, CH-4051 Basel, Switzerland.}

\email{\textcolor[rgb]{0.00,0.00,0.84}{catherine.bandle@unibas.ch}}



\dedicatory{This paper is dedicated to Professor J. E. Pecaric\\
\vspace{.5cm} {\rm Submitted by P. K. Sahoo}}

\subjclass[2000]{Primary 35P15, 47A75 ; Secondary 49R50, 51M16.}

\keywords{Estimates of eigenvalues, trace inequality, comparison
theorems for eigenvalues.}

\date{Received: 21 April 2008; Accepted 20 May 2008.}

\begin{abstract}
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 Bandle, v. Bellow and Reichel
in [J. Eur. Math. Soc., 10 (2007), 73--104]. We then apply these
bounds to the trace inequality.

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