A vectorial problem with thin free boundary

Abstract We consider the vectorial analogue of thin free boundary problem introduced by Caffarelli et al. (J Eur math Soc 12:1151-1179, 2010) as a realization nonlocal version classical Bernoulli problem. study optimal regularity, nondegeneracy, and density properties local minimizers. Via blow-up analysis based on Weiss type monotonicity formula, we show that is union “regular” “singular” part. Finally use viscosity approach to prove $$C^{1,\alpha }$$ C 1 , α regularity regular part boundary.