A variational approach to parabolic equations

Parabolic partial differential equations are an active field of research, both from theoretical and applied point of view. In the literature, several existence and uniqueness results concerning global solutions of the weak formulation have been established. After presenting some of these classical results, we will focus on a particular kind of PDE, i.e., the time-dependent version of the Euler-Lagrange equation. We will discuss a recent result concerning the existence of global weak solution to this problem, obtained through a variational approach. In particular, a notion of minimizer can be introduced also in the parabolic setting; precisely, in some cases, a function is a solution of a parabolic equation if and only if it satisfies a certain variational inequality. Hence, the idea is to recast the original problem in a different setting which allows to look for such minimizers. Then, one can introduce suitable functionals whose (formal) Euler-Lagrange equation is an approximation of the original PDE, and use the methods from the calculus of variations to obtain minimizers, which converge (through a suitable limit procedure) to the required parabolic minimizer.