Analytic solutions for the approximated Kantorovich mass transfer problems by $p$-Laplacian approach

TL;DR

This paper presents an analytic method using $p$-Laplacian equations and canonical duality theory to approximate and solve the Kantorovich mass transfer problem, demonstrating convergence to the global maximizer.

Contribution

It introduces a novel $p$-Laplacian based approach combined with canonical duality theory to analytically approximate solutions for the Kantorovich mass transfer problem.

Findings

Derived a sequence of analytic solutions for the minimization problems

Proved convergence of the solutions to the global maximizer

Established a new approximation framework for mass transfer problems

Abstract

This manuscript discusses the approximation of a global maximizer of the Kantorovich mass transfer problem through the approach of $p$-Laplacian equation. 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 Kantorovich problem will be demonstrated.