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Communications in Mathematical Physics

, Volume 271, Issue 1, pp 93–101 | Cite as

The Singular Set for the Composite Membrane Problem

  • Henrik ShahgholianEmail author
Article

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

Keywords

Composite Membrane Free Boundary Problem Optimal Pair Monotonicity Formula Density Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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