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 .

April 23, 2013 at 9:37 pmNumerical Approximation using Relaxed Formulation  Beni Bogoşel's blog