My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs?
Honestly, my understanding of $\Gamma$-convergence is that relatively vague: given a sequence of functionals we expect the limit to have an optimal (in some sense) lower bound, which is common for every element in the sequence.
Now, writing a PDE in the weak formulation apparently gives us ability to apply functional-related tools to it. I heard people saying that for elliptic PDEs $\Gamma$-convergence basically establishes convergence of minimizers of corresponding functionals, whereas for hyperbolic PDEs it corresponds to minimizing gradient flow.
Do the statements I provided in above paragraph have any reasonable and intuitive justifications, or is it all just nonsense? If they are true, what would be the similar statement for parabolic equations?