We study the minimum problem for the functional with the constraint for , where is a bounded domain and . First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary .
"A minimization problem with free boundary related to a cooperative system." Duke Math. J. 167 (10) 1825 - 1882, 15 July 2018. https://doi.org/10.1215/00127094-2018-0007