  In this paper, we discuss adaptive approximations of an elliptic eigenvalue
optimization problem in a phase-field setting by a conforming finite element
method. An adaptive algorithm is proposed and implemented in several two
dimensional numerical examples for illustration of efficiency and accuracy.
Theoretical findings consist in the vanishing limit of a subsequence of
estimators and the convergence of the relevant subsequence of
adaptively-generated solutions to a solution to the continuous optimality
system.
