Ana, 14, was murdered in May 2018 by two boys who were 13 at the time.
Her body was found in a disused house in south-west Dublin.
Mr Justice Michael White said the publication of anything identifying the two boys, was a breach of the Children Act and of the court’s orders.
He said no individual had the right to put anything on social media identifying or tending to identify the teenagers.
The court also allowed the DPP to serve notice on the two companies that she intends to seek orders for the attachment of their assets and the committal to prison of those responsible for publishing the material.
That application will be back before the court on Thursday.
A spokesperson for Facebook said: “As soon as we became aware of content identifying Boy A and Boy B being shared on Facebook this morning, we removed this content immediately for violating our community standards and local law.
“We also applied our photo-matching technology to prevent this content from being re-shared on Facebook, Instagram or Messenger. We will continue to remove this content from our platforms.”
Ana was adopted from Russia when she was two by Irish woman Geraldine Kriégel and her French-born husband Patric.
The Ceann Comhairle (speaker) intervened to remind members that the case was still before the courts as the two boys await sentencing in July.
Leo Varadkar said it was too early to know if UK pornography legislation was effective
Mr Varadkar said it was “a matter of concern that pornography is so accessible” and that young people learn about sex through pornography.
“That’s not healthy,” he added.
He said the UK law was relatively new and that it was too early to know whether it had been effective before deciding whether to introduce similar legislation in the Republic of Ireland.