The singular set for the composite membrane problem

Abstract. In this paper we study the behavior of the singular set {u = |∇u | = 0}, for solutions u to the free boundary problem ∆u = fχ{u≥0} − gχ{u<0}, with f(x)+ g(x) < 0, f, g ∈W 1,p ∩C0. Such problems arises in an eigenvalue optimization for composite membranes. Here we show that if for a singular point z ∈ {u = ∇u = 0}, the density is positive |{u ≥ 0} ∩Br(z) | ≥ c0rn, for some c0> 0, then z is isolated. The density assumption can be motivated by the following example u = x21, f = 2, g < −2, and {u < 0} = ∅