The Singular Set for the Composite Membrane Problem

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 > 0, f(x) + g(x) < 0, and f, g ∈ Cα. Such problems arises in an eigenvalue optimization for composite membranes. Here we show that if for a singular point z ∈ {u = ∇u = 0}, there are r0 > 0, and c0 > 0 such that the density assumption |{u < 0} ∩Br(z)| ≥ c0r, ∀ r < r0, holds, then z is isolated. The density assumption can be motivated by the following example u = x1, f = 2, g < −2, and {u < 0} = ∅.