Applications of Optimization to the Nonlinear Differential Equations

Question

I am really interested in the potential applications of optimizations (nonlinear, convex, and integer programming) and the functional analysis (function spaces and nonlinear operator theory) to the study of differential equations. Basically, I am interested in their methodological applications to solve DEs. By any chance, could you recommend me good nonlinear differential equations where optimization techniques and functional analysis could be applied well? Strangely, I have not found any nonlinear DE that fit the criteria..

Answer 1

Calculus of variations studies the solutions of certain differential equations (known as variational) as minimizers of associated cost functionals. For example, the minimizers of the function graph area functional $u\mapsto\int_D\sqrt{1+|\nabla u|^2}$, for a two-dimensional bounded domain $D$ are solutions of the nonlinear partial differential equation $\nabla\cdot[\nabla u/\sqrt{1+|\nabla u|^2}]=0$ on $D$. Prescribing boundary values for $u$ that are not constant leads to nontrivial solutions.