  This paper addresses the geometric optimization problem of the first Robin
eigenvalue in exterior domains, specifically the lowest point of the spectrum
of the Laplace operator under Robin boundary conditions in the complement of a
bounded domain. In contrast to the Laplace operator on bounded domains, the
spectrum of this operator is not purely discrete. The discrete nature of the
first eigenvalue depends on the parameter of the Robin boundary condition. In
two dimensions, D. Krejcirik and V. Lotoreichik show that the ball maximizes
the first Robin eigenvalue among all smooth, bounded, simply connected sets
with given perimeter or given area, provided the eigenvalue is discrete.
  We extend these findings to higher dimensions. The discrete spectrum of the
Laplace operator under Robin boundary conditions can be characterized through
the Steklov eigenvalue problem in exterior domains, a topic studied by G.
Auchmuty and Q. Han. Assuming that the lowest point of the spectrum is a
discrete eigenvalue, we show that the ball is a local maximizer among nearly
spherical domains with prescribed measure. However, in general, the ball does
not emerge as the global maximizer for the first Robin eigenvalue under either
prescribed measure or prescribed perimeter.
