Abstract
We show that a certain eigenvalue minimization problem in
two dimensions for the Laplace operator in conformal classes
is equivalent to the composite membrane problem. We again
establish such a link in higher dimensions for eigenvalue problems
stemming from the critical GJMS operators. New free
boundary problems of unstable type arise in higher dimensions
linked to the critical GJMS operator. In dimension four,
the critical GJMS operator is exactly the Paneitz operator.