The talk deals with the minimax optimization of the energy 
corresponding to the solution of an elliptic equation. The 
maximum of the energy is chosen over a constrained class of 
boundary functions (Neumann data on the boundary). Then, the 
minimum is taken over the coefficients of the equation modeling 
the material properties in the domain. The L^2 constraint for 
the class of boundary functions leads to optimization of the 
Steklov eigenvalues. Symmetries of the optimal solution and 
bifurcations are characteristic features of the problem. 
It is shown that an invariance of the constraints under a 
symmetry transformation leads to a symmetry of the optimal 
solution. Continuous variation of the constraints may cause 
bifurcation of the solution. Problems of this type arise in 
inverse imaging and structural design.