SEPARATION OF A LOWER DIMENSIONAL FREE BOUNDARY IN A TWO-PHASE PROBLEM

We study minimizers of the energy functional integral(D) vertical bar x(n)vertical bar(a)vertical bar del u vertical bar(2) + integral(D boolean AND(Rn-1 x{0})) lambda(+) chi({u>0}) + lambda(-) chi({u<0}) dH(n-1) without any sign restriction on the function u. The main result states that the two free boundaries Gamma(+) = partial derivative{u( . , 0) > 0} and Gamma(-) = partial derivative{u( . , 0) < 0} cannot touch, i.e., Gamma(+) boolean AND Gamma(-) = theta.