Quasiconformal mappings as a tool to study certain two-dimensional \(G\)-closure problems (Q1919684)

A recent theorem due to Astala establishes the best exponent for the area distortion of planar \(K\)-quasiconformal mappings. The author uses a refinement of Astala's theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number \(n\) of possibly anisotropic phases in prescribed volume fractions. For \(n = 2\) he proves the optimality of the bounds under certain additional assumptions on the \(G\)-closure parameters.