The Singular Set for the Composite Membrane Problem

Abstract

In this paper we study the behavior of the singular set

$$ \{u=|\nabla u| =0\}, $$

for solutions u to the free boundary problem

$$ \Delta u = f\chi_{\{u\geq 0\} } -g\chi_{\{u < 0\}},$$

with $f > 0$, f(x) + g(x) < 0, and $f,g \in C^\alpha$. Such problems arise in an eigenvalue optimization for composite membranes. Here we show that if for a singular point $z\in \{u=\nabla u=0\}$, there are r ₀ > 0, and c ₀ > 0 such that the density assumption

$$ |\{u < 0\}\cap B_r(z)|\geq c_0r^2, \qquad \forall \ r < r_0,$$

holds, then z is isolated. The density assumption can be motivated by the following example:

$$ u=x_1^2, \quad f=2, \quad g < -2, \quad \hbox{and } \{u < 0\}=\emptyset.$$