$\Gamma$-convergence (Gamma-convergence) and PDEs?

Question

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?

Answer 1

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.