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Multiple solutions for a nonconvex variational problem with double‐well potentials
Author(s) -
Lu Xiaojun,
Lv Xiaofen,
Wen Guanghui
Publication year - 2018
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2434
Subject(s) - mathematics , maxima and minima , duality (order theory) , boundary value problem , nonlinear system , algebraic number , euler's formula , algebraic equation , mathematical analysis , pure mathematics , physics , quantum mechanics
Summary In this paper, we present a set of complete solutions for a nonconvex variational problem with double‐well potentials in higher dimensions. Based on the canonical duality theory, the corresponding nonlinear Euler‐Lagrange equation with Neumann boundary condition can be converted into an algebraic equation, which can be solved analytically to obtain the solutions of the dual problem. Correspondingly, local extrema of the primal problem can be identified by the pure complementary energy principle.