Analytic solutions for the approximation of $p$-Laplacian problem

TL;DR

This paper develops an analytical approach using canonical duality theory to approximate solutions for the nonlinear p-Laplacian problem, transforming it into a sequence of solvable minimization and maximization problems.

Contribution

It introduces a novel approximation mechanism and applies canonical duality theory to analytically solve the p-Laplacian problem, including global extrema analysis.

Findings

Sequence of analytic minimizers obtained

Global extrema for primal and dual problems identified

Transformation simplifies nonlinear PDE into dual problems

Abstract

This paper mainly investigates the analytic solutions for the approximation of $p$-Laplacian problem. Through an approximation mechanism, we convert the nonlinear partial differential equation with Dirichlet boundary into a sequence of minimization problems. And a sequence of analytic minimizers can be obtained by applying the canonical duality theory. Moreover, the nonlinear canonical transformation gives a sequence of perfect dual maximization(minimization) problems, and further discussion shows the global extrema for both primal and dual problems.