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Multi‐Duality in Minimal Surface—Type Problems
Author(s) -
Gao David Yang,
Yang Wei H.
Publication year - 1995
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1995952127
Subject(s) - mathematics , duality (order theory) , type (biology) , minimal surface , affine transformation , operator (biology) , nonlinear system , curvature , dual (grammatical number) , mean curvature , mathematical analysis , mathematical optimization , pure mathematics , geometry , physics , art , ecology , biochemistry , chemistry , literature , repressor , quantum mechanics , gene , transcription factor , biology
The multi‐duality of the nonlinear variational problem inf J ( u , Λ 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 Λ. 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.