Gradient estimates for solutions to the conductivity problem

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Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity (2012, background): “In this relation we mention that a precise dependence on k when D1 and D2 are disks was shown in [3,5].”
On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D (2013, background): “Further, we note that if w is a nontrivial solution of the homogeneous equation (14) for k = 1/β, then the harmonic conjugate w̃ of w (see for instance the construction in [5], Section 3) satisfies { div ( [1 + (k̃ − 1)1D(x)]∇w̃(x) ) = 0 in R2 w̃(x) ∈ W 1,−1 0 (R2).”