Generalizations of wave equations to multidimensional variational problems

This is a survey paper based on previous results of the author. In the paper, we define and discuss the generalizations of linear partial differential equations to multidimensional variational problems. We consider two examples of such equations: first, the generalized Schr"odinger equation which is a natural candidate for the mathematical equation of quantum field theory, and second, the quantum Plato problem which is a natural candidate for a simplest mathematical equation of string theory and, more generally, theory of $D$-branes. We propose a way to give a mathematical sense to these equations.