On the extrema of a nonconvex functional with double-well potential in higher dimensions

TL;DR

This paper investigates the local extrema of a nonconvex functional with double-well potential in higher dimensions using the canonical duality method, revealing significant differences from the one-dimensional case.

Contribution

It extends the analysis of nonconvex functionals with double-well potentials to higher dimensions and introduces a dual algebraic approach to find local extrema.

Findings

Higher-dimensional extrema differ significantly from 1D cases.

The canonical duality method simplifies the Euler-Lagrange equation to a cubic algebraic equation.

The approach provides a new way to analyze nonconvex functionals in multiple dimensions.

Abstract

This paper mainly addresses the extrema of a nonconvex functional with double-well potential in higher dimensions through the approach of nonlinear partial differential equations. Based on the canonical duality method, the corresponding Euler--Lagrange equation with Neumann boundary condition can be converted into a cubic dual algebraic equation, which will help find the local extrema for the primal problem. In comparison with the 1D case discussed by D. Gao and R. Ogden, there exists huge difference in higher dimensions, which will be explained in the theorem.