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Mulligan scored in May's 3-0 European Championship win over Norway for Ireland's Under 19 side - but unfortunately Sean Caffrey's side failed to progress to the final stages.
Elliott's withdrawal is a serious blow as he is the Under 21s record goalscorer.
"It is the same niggle I have been carrying since pre-season but hopefully this will put an end to it," said Elliott to the Black Cats' website. "It is disappointing to miss out on the international game next week but I have spoken to Mick McCarthy about it and also to Don (Givens) and they are both fine about it."
A regular for the Wolves reserves, Mulligan has made one appearance as a substitute in the 2004/05 Championship for boss Dave Jones.
Given's Republic side could do with a win but are already playing catch up against first-placed Switzerland and France - their hosts in Troyes on Friday (kick-off 8pm, Irish time).
Glenn Whelan, John Fitzgerald and Kevin Doyle all scored in last month's opening 3-0 Group 4 win over Cyprus at Kilkenny City's Buckley Park. But a poor performance in Bern four days later (September 7) saw the Irish lose 4-2 to the excellent Swiss.
Elliott - scoring his sixth in ten appearances - brought the tie back to 3-1 just after the hour, but an own goal from Paddy McCarthy spelt the end for Givens' charges. Celtic midfielder Aiden McGeady earnt scant consolation with an 86th minute penalty.
France meanwhile, have played just once so far and thanks to a lone Anthony Le Tallec goal against Israel last month, have a 100% record. Le Tallec is on loan at Saint-Etienne from Liverpool.
It was new coach Rene Girard's first game in charge since former U21 boss Raymon Domenech was made senior coach of Le Bleus.