Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications (Q1586659)

From author's \textit{Problems and motivations}: ``We are interested in the general analytic solution of the nonconvex variational problem \[ \min P(u)=\int_{I}W(\Lambda (u)) dx- \int_{I}fu dx, \quad u\in {\mathcal U}_{a},\tag \(\mathcal P\) \] where \(I\subset R\) is an open interval, \(f(x)\) is a given function, \(\Lambda \) is a nonlinear operator, and \(W(\varepsilon)\in {\mathcal L}(I)\) is a piecewise Gâteaux differentiable function of \(\varepsilon =\Lambda (u);\) \({\mathcal U}_{a}\) is a closed convex subspace of a reflexive Banach space \( {\mathcal U}.\) Duality theory for general fully nonlinear, nonsmooth variational problems was originally studied by Gao and Strang for \(n\)-dimensional finite deformation systems. In order to recover the duality gap between the primal and dual variational problems, a so-called complementary gap function was discovered. A general complementary variational principle for fully nonlinear, nonsmooth systems was proposed. Applications of this general theory have been given in a series of publications on finite deformation mechanics. An interesting triality theory was proposed and a so-called nonlinear dual transformation method has been developed for solving general nonconvex variational/boundary value problems. In the present paper, this nonlinear dual transformation method is generalized for obtaining the general analytic solutions of the nonconvex, nonsmooth variational problem \(({\mathcal P})\)''.