I will describe a scheme for converting the question of solving a given system of equations and associated side-conditions into a variational problem whose formal Euler-Lagrange system recovers the given system and side-conditions using a change-of-variables generated by the scheme. The variational principle is convex, and the method uses a string of such convex variational principles to seek a solution of the original question as an interior minimizer of the last variational problem of the string. Weak coercivity of a variational problem in the string endows its minimizer the status of a weaker `variational dual solution’ to the original, possibly non-variational problem. This last point of view is directly related to Y. Brenier’s Hidden Convexity approach to nonlinear PDE.
Due to the inherent convexity of the scheme, it allows for obtaining `unstable’ solutions of the original, primal system of equations in a `stable’ manner. A ChatGPT-enabled exploration of the scheme in the context of approximation of the mixed curvature isometric embedding problem from R^2 to R^3 will be discussed, as well as the problem of finding deformations of a membrane with prescribed principal stretch fields. This last problem is practically important for questions of sensing and actuation.