Poincaré’s Variational Problem in Potential Theory

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Surface Plasmon Resonance of Nanoparticles and Applications in Imaging (2014, unknown): “If a and b denote the semi-axis lengths of an ellipse then it can be shown that ±((a − b)/(a + b))i are its Fredholm eigenvalues [44].”
Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity (2012, background): “It is proved in [1] based on a result in [16] that −S is positive semi-definite and self-adjoint, SK∗ is self-adjoint, and, hence, there is a self-adjoint operator A on H such that √−SK∗ = A√−S.”
Mathematical Analysis of Plasmonic Nanoparticles: The Scalar Case (2015, background): “On the other hand, it is known that in H∗(∂D) [27] σ(K∗ D)\{1/2} = { ±(1) 2 ( a− b a+ b )j , j = 1, 2, .”
Spectral bounds for the Neumann-Poincaré operator on planar domains with corners (2012, unknown): “Repeating the calculations of ([11], Lemma 5) verbatim, we have for f ∈ H Considering the unique decomposition h = Df + Sg of any h ∈ H i , the result follows via the unitary identiﬁcation H i (cid:51) ( h i , 0) (cid:55)→ ∇ h i ∈ B (Ω).”
Some Historical Remarks on the Positivity of Boundary Integral Operators (2007, methodology): “Poincar´e’s own estimates are being revisited and adapted to a modern standard in the very recent paper [ 18 ] which treats the same framework as Poincar´e did, namely two- and three-dimensional potential theory on smooth domains.”