Variational Convex Analysis: Applications to Non-convex Models

This work develops duality for systems originally modeled by non-linear differential equations. For such problems, the variational formulations are in general non-convex. In many situations the primal approaches have no solutions in the classical sense, being the minimizing sequences only weakly convergent. However, through the dual formulations, it is possible to compute these weak limits, and such evaluations have many practical applications, such as for composites in elasticity, phase transition models, problems in micro-magnetism, and others. Among such results in variational analysis, we also highlight the establishment of a linear system whose the solution also solves the incompressible Navier-Stokes system. To summarize, we introduce convex analysis as an interesting alternative tool for the undestanding and computation of some important problems in the calculus of variations.