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?
This is a vague question and I can sadly only give a vague answer in a few sentences....$\Gamma$-convergence is a very powerful tool when it comes to a specific type of problems that are obtained as limits of certain types of other problems. One formulates solutions of PDEs as critical points of specific functionals. $\Gamma$-convergence of these functionals is a tool to obtain a limit functional and also convergence of critical points to a critical point of this limit functional.
Now for more specific examples try Andrea Braides' book "$\Gamma$-convergence for beginners": it has a very nice introduction with many examples.