On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D

We study the spectrum of the Poincaré variational problem for two close to touching inclusions in R2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.

[1]  Biao Yin,et al.  Gradient Estimates for the Perfect Conductivity Problem , 2006 .

[2]  Eric Bonnetier,et al.  An Elliptic Regularity Result for a Composite Medium with "Touching" Fibers of Circular Cross-Section , 2000, SIAM J. Math. Anal..

[3]  Habib Ammari,et al.  Gradient estimates for solutions to the conductivity problem , 2005 .

[4]  Habib Ammari,et al.  Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions , 2009 .

[5]  Ivo Babuška,et al.  Damage analysis of fiber composites Part I: Statistical analysis on fiber scale , 1999 .

[6]  H. Shapiro,et al.  Poincaré’s Variational Problem in Potential Theory , 2007 .

[7]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[8]  M. Krein.  Compact linear operators on functional spaces with two norms , 1998 .

[9]  H. Ammari,et al.  Estimates for the electric field in the presence of adjacent perfectly conducting spheres , 2007 .

[10]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[11]  KiHyun Yun,et al.  Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections ✩ , 2009 .

[12]  A generalized eigenproblem for the Laplacian which arises in lightning , 2008 .

[13]  Habib Ammari,et al.  Spectral Theory of a Neumann–Poincaré-Type Operator and Analysis of Cloaking Due to Anomalous Localized Resonance , 2011, 1212.5066.

[14]  Henri Poincaré.  La méthode de Neumann et le problème de Dirichlet , 1897 .

[15]  H. Ammari,et al.  Optimal estimates for the electric field in two dimensions , 2006 .

[16]  M. Vogelius,et al.  Gradient Estimates for Solutions to Divergence Form Elliptic Equations with Discontinuous Coefficients , 2000 .

[17]  George F. Carrier,et al.  High Shear Stresses in Stiff-Fiber Composites , 1984 .

[18]  O. Kirichenko,et al.  Electromagnetic and acoustic waves in layered organic conductors (a review) , 1999 .

[19]  J. Schauder,et al.  Potentialtheoretische Untersuchungen , 1931 .

[20]  V. Maz'ya.  Functions with a finite dirichlet integral in a domain with cusp points on the boundary , 1984 .

[21]  Eric Bonnetier,et al.  Pointwise bounds on the gradient and the spectrum of the Neumann--Poincare operator: The case of 2 discs , 2011 .

[22]  Xanthippi Markenscoff,et al.  Stress amplification in vanishingly small geometries , 1996 .

[23]  M. Lim,et al.  Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities , 2011, 1105.4328.

[24]  Louis Nirenberg,et al.  Estimates for elliptic systems from composite material , 2003 .

[25]  Martin Costabel,et al.  Some Historical Remarks on the Positivity of Boundary Integral Operators , 2007 .