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