As far as I can see what you are trying to do is the following
- Connect the monitor to your Mac mini to display video
- and also send audio from the Mac mini to the same monitors built-in speakers
As DisplayPort supports running both video and audio via the same single cable this is the obvious connection to use on the monitor.
However you are going the wrong route trying to go from HDMI on the Mac to DisplayPort on the monitor. As should from the other answers be now obvious this would require a far more complicated (and expensive) sort of adapter.
What you should instead be doing is using a far more simple (and much cheaper) Mini DisplayPort to DisplayPort cable. The Apple Mini DisplayPort connector is 100% compatible signalwise with a bigger DisplayPort connector.
See http://vimeo.com/5000770
See also http://www.displayport.org/consumer/?q=content/faq
and read in particular
A: Standard DisplayPort-to-Mini DisplayPort adaptors and cables work both ways between sources and sinks:
- Mini DisplayortP graphics card to standard DisplayPort monitor
- Standard DisplayPort graphics card to mini-DisPlayPort monitor
You can buy a suitable cable from here http://www.amazon.com/DisplayPort-32AWG-Cable-Plated-Connectors/dp/B0034X6SCY/re f=sr_1_3?ie=UTF8&qid=1314366162&sr=8-3
Once this is all connected up, it should merely be a matter of going to System Preferences, then Sound, and telling the Mac to send sound via its Mini DisplayPort connection.