# The Singular Set for the Composite Membrane Problem

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## Abstract

In this paper we study the behavior of the singular set
for solutions with \(f > 0\), holds, then

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

*u*to the free boundary problem$$ \Delta u = f\chi_{\{u\geq 0\} } -g\chi_{\{u < 0\}},$$

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

*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
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## References

- 1.Alt H.W., Caffarelli L.A. and Friedman A. (1984). Variational problems with two phases and their free boundaries.
*Trans. Amer. Math. Soc.*282: 431–461 CrossRefMathSciNetGoogle Scholar - 2.Andersson, J., Weiss, G.S.: Cross-shaped and degenarte singularities in an unstable elliptic free boundary problem. J. Differ. Eqs. 2005. In press.Google Scholar
- 3.Blank I. (2004). Eliminating mixed asymptotics in obstacle type free boundary problems.
*Commun. Partial Differ. Eq.*29(7–8): 1167–1186 zbMATHCrossRefMathSciNetGoogle Scholar - 4.Caffarelli, L.A., Jerison, D., Kenig, C.E.: Some new monotonicity theorems with applications to free boundary problems. Ann. Math. (2)
**155**(2) 369–404 (2002)Google Scholar - 5.Chanillo, S., Grieser, D., Kurata, K.: The free boundary problem in the optimization of composite membranes. In:
*Differential geometric methods in the control of partial differential equations (Boulder, CO*, 1999), Contemp. Math.**268**, Providence, RI: Amer. Math. Soc. 2000, pp. 61–81Google Scholar - 6.Chanillo S., Grieser D., Imai M., Kurata K. and Ohnishi I. (2000). Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes.
*Commun. Math. Phys.*214(2): 315–337 zbMATHCrossRefMathSciNetADSGoogle Scholar - 7.Petrosyan, A., Shahgholian, H.: Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem. Submitted, available at http://www.math.purdue.edu/ arshak/pdf/gecm-energy-final.pdfGoogle Scholar
- 8.Weiss G.S. (2001). An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary.
*Interfaces Free Bound.*3(2): 121–128 zbMATHMathSciNetCrossRefGoogle Scholar - 9.Shahgholian H., Uraltseva N. and Weiss G.S. (2004). Global solutions of an obstacle-problem-like equation with two phases.
*Monatsh Math*142(12): 27–34 zbMATHCrossRefMathSciNetGoogle Scholar - 10.Shahgholian, H., Uraltseva, N., Weiss, G.S.: The Two-Phase Membrane Problem – Regularity of the Free Boundaries in Higher Dimensions. Submitted, available at http://www.ms.u-tokyo.ac.jp/ gw/url.pdfGoogle Scholar

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© Springer-Verlag 2006