MULTI-DUALITY IN MINIMAL SURFACE-TYPE PROBLEMS

The multi-duality of the nonlinear variational problem inf J(u,Lambda u) is studied for minimal surfaces-type problems. By using the method developed by Gao and Strang [1], the Fenchel-Rockafellar's duality theory is generalized to the problems with affine operator Lambda, Two dual variational principles are established for nonparametric surfaces with constant mean curvature. We show that for the same primal problem, there may exist different dual problems. The primal problem may or may not possess a solution, whereas each dual problem possesses a unique solution. An evolutionary method for solving the nonlinear optimal-shape design problem is presented with numerical results.