## The Basic properties of Gamma Convergence

Let be a metric space, and for let be given . We say that -converges to on as , and we write , if the following conditions hold:

**(LI)** For every and every sequence such that in we have

**(LS)** For every there exists a sequence such that in and

The -convergence has the following properties:

1. The -limit is always lower semicontinuous on ;

2. -convergence is stable under continuous perturbations: if and is continuous, then

3. If and minimizes over , then every limit point of minimizes over .

I have seen these properties stated in many places, but the proofs are usually left to the reader. I will try and give the proofs below.

**Proofs** 1. Take a sequence in such that in and . Assume that there exists for which we have .

The properties of the convergence combined prove that for every there exists a sequence in such that .

We apply this result to each of the above and we find that there exists a sequence such that and . Obviously we have and by the property of the convergence

On the other hand for great enough we have and as a consequence for great enough we have . We have reached a contradiction. Therefore we have

and is lower semicontinuous.

2. It is easy to see that verifies both with equality, for every sequence , thus proving the desired result immediatley by using the properties of for the sum of two sequences.

3. Suppose there exists and such that . Take another point and the recovery sequence (which exists as a consequence of the -convergence) such that

Then we have

proving that . Since this is true for every we find that is a minimizer for .

please can you help me to prouve that $q(x)\leq \liminf_{n\rightarrow +\infty}q_n(x_n)$

where $q(x)=\,$, $q_n(x_n)=\,$, A an operator in Hilbert space H ans $A_n$ the approximation Yosida of A and (x_n) sequence witch converge to x in H

( Iam french)

Sorry, but I have no idea how to help you on this one… You should write an exact description of your problem. I don’t even know what is a Yosida approximation…