Okay, good. Well, so today we continue with gamma convergence and gradient theory of phase

transition. So, this is the

second lecture, second lecture on this topic, while this is actually the 11th

lecture on Stoudon, if I'm not wrong. Okay, so we started with, so last time we

started with the theory of gamma convergence in particular, we started, so

we analyzed why is gamma convergence important and what is the notion of gamma

convergence. Let me just remind you the definition. Remember that we have x, d, a

metric space, we have a sequence of functionals, f epsilon and another

functional which is, should be the limit. Okay, so we say that f of x is the gamma

limit as epsilon goes to zero of f epsilon at x. Here we are basically, this

is a, we are evaluating at x, so is the gamma limit at a point x in the space. If

and only if we have one, the limit in equality, which is for every sequence x

epsilon such that x epsilon converges to x, we have the usual limit in equality,

okay, for every sequence, plus the, actually the Liem's inequality or we

also, we also mentioned that this is, I mean this is the Liem's inequality or

the existence of a recovering sequence, which means that there exists a, there

exists a sequence at x bar epsilon such that x bar epsilon is converging to x as

epsilon goes to zero and we have the Liem's inequality. Okay, I mean I've also seen

that basically this condition here, so sometimes it's called existence of a

recovery sequence or we also consider the approximate Liem's inequality. Anyhow,

so this is the definition of gamma limit. Of course, if these old for, these old for

all x, then we basically have that, so if for all x, then f is the gamma limit. Now,

so last time we have seen actually that you are interested, not only in, so the

gamma limit is a good notion for the asymptotic behavior of a sequence of

minimization problems. In particular, we have seen that, so our purpose is we have,

I don't know, a sequence of minimization problem, epsilon x such that x, and

basically we would like to understand what happens when epsilon goes to zero

without passing through the properties of the minimizers. Okay, and so somehow, so

basically we would like to be able to, to be able to build a link between these two

problems as epsilon goes to zero, and in particular, so we either, so basically we

always want to, so either we want to approximate, so this problem, okay,

basically meaning that we are going from this minimization problem to the other

one, or for instance, sometimes we can also do the other way around, for instance,

when this problem is difficult to solve, we may, I don't know, we may define a

sequence of functional for which this minimization problem is easier, and then

basically we want to pass to the limit as epsilon goes to zero. We have seen

that what is very important is, so the main theorem of gamma convergence is

that equicorpsimness plus gamma convergence implies convergence of

the minimization problem, problem which is basically that there exists a minimum

over X of f which is namely equal to the limit of the inf of f epsilon, where the

equicorpsimness is very important because it tells us that we can actually,

that there exists a pre-compact minimizing sequence, okay, which is very

important because, so the idea behind, so this result is actually to, so comes from

the direct method of calculus operation, okay, so equicorpsimness means that the

sublevels of the sequence of the functionals are contained in a compact

set contained in X, okay, which in particular means, I remind you that there exists a

minimizing sequence, there exists X epsilon such that X epsilon, a

subsequence converges to some X, I don't know, and limit as K of f epsilon K, epsilon K is actually the

limit, I'll show you the limit, yeah, okay, so basically here we are just stating that

there exists a minimizing sequence, there exists a minimizing sequence which is