  We consider smooth bounded surfaces with a smooth boundary and a prescribed
background metric g_0. We now consider all metrics g conformal to g_0 which
have a prescribed volume M. We now minimize the first eigenvalue of the Laplace
operator of g over the metrics conformal to g_0 and having the prescribed
volume. We show that this problem is equivalent to the study of the Composite
Membrane Problem, a free boundary problem studied earlier by the author and his
collaborators in all dimensions. Thus complete answers, existence of the limit
metric, regularity of the minimizing eigenfunction and various qualitative
properties of the metric are easily obtained from the solution of the Composite
Membrane problem. The problem of minimizing eigenvalues over conformal classes
has a higher dimensional analog for the critical GJMS operator and leads to new
classes and questions for higher order unstable free boundary problems. In
particular in dimension 4 we are lead to free boundary problems involving the
Paneitz operator and in odd dimensions to fractional free boundary problems of
unstable type.
