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THREE SINGULAR VARIATIONAL PROBLEMS (Viscosity Solutions of Differential Equations and Related Topics)

This paper mainly investigates the approximation of a global maximizer of the 1-D Monge–Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Monge–Kantorovich problem will be demonstrated.

Analytic solutions for the approximated 1-D Monge–Kantorovich mass transfer problems

This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational method. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the duality method gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of optimal mapping for the Monge problem according to the theoretical results.

Xiaojun Lu

Papers

Canonical duality approach in the approximation of optimal Monge mass transfer mapping