Multiple nonsmooth solutions for nonconvex variational boundary value problems in $\mathbb{R}^n$

TL;DR

This paper utilizes canonical duality-triality theory to analytically solve a nonconvex variational problem with a double-well potential, revealing multiple solutions and extremality conditions in boundary value problems.

Contribution

It introduces a method to convert complex nonlinear differential equations into algebraic equations for complete solution sets using duality theory.

Findings

All solutions obtained analytically in the dual space.

Identification of global and local extremality conditions.

Application to mechanical models in two-dimensional space.

Abstract

This paper presents a set of complete solutions of a nonconvex variational problem with a double-well potential. Based on the canonical duality-triality theory, the associated nonlinear differential equation with either Dirichlet/Neumann or mixed boundary conditions can be converted into an algebraic equation, which can be solved analytically to obtain all solutions in the dual space. Both global and local extremality conditions are identified by the triality theory. In the application part, typical mechanical models with specificsources and boundary conditions in $\mathbb{R}^2$ are exhibited.