I signed a lease in May for a rental to commence in Sept.
I gave owner a check for Sept. rent and a check for security. Signed lease, did not get copy back.
Due to a sudden change in marital status, I was not able to move in, gave 3 weeks notice via e mail. Did not have owners address or telephone.. had to Google them.
They rented the apartment and want to penalize me and not return my money in full.

They did not send any bank info about the deposit either. Do I have a case..

First, without knowing where you are we can't tell much. It really amazes me that people think laws are universal and it wouldn't matter where they are located.

Second, what grounds are they citing for not returning the monies in full?

Third, when did they rent the apartment with respect to when you moved in?

Fourth, you said you gave 3 weeks notice, I'm assuming this was 3 weeks prior to your move in date.

Please do not ask questions via e-mail or PM. Any followups should be asked in the thread. You really can't expect us to give a legal answer without detail.

If you gave only 3 weeks notice prior to move in date, then their witholding a small amount for the costs of re renting is reasonable.