There is no open source license as far as I can tell. README on Elixir’s Github repo states:
“Elixir” and the Elixir logo are copyright © 2012 Plataformatec.
and I did receive answer from @josevalim here back in a day:
Generally, I think you should get in touch with Jose or someone at Plataformatec for any usage of logo, comercially or otherwise.
And when it comes to name - I do not believe the statement in README that “Elixir” is copyrighted by Plataformatec true. It does not bear the required markings that it’s copyrighted on any web site or materials I saw. Most likely, according to your local law, you can use “Elixir” term in any context related to the language freely. If Plataformatec were to register it as a trademark, I can imagine the process being lengthy and very much costly, especially that the term is pretty popular and copyrighted as atrademark by other holders worldwide.