Composites and quasiconformal mappings: new optimal bounds in two dimensions

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We prove new bounds for the homogenized tensor of two dimensional multiphase conducting composites. The bounds are optimal for a large class of composites. In physical terms these are mixtures of one polycrystal and several isotropic phases, with prescribed volume fractions. Optimality is understood in the strongest possible sense of exact microgeometries. The techniques to prove the bounds for composites are based on variational methods and results from quasiconformal mappings. We need to refine the quasiconformal area distortion theorem due to the first author and prove new distortion results with weigths. These distortion theorems are of independent interest for PDE’s and quasiconformal mappings. They imply e.g. the following surprising theorem on integrability of derivatives at the borderline case: For K > 1, if \(f\in W^{1,2}_{\rm loc}({\bf R}^2,{\bf R}^2)\) is K-quasiregular, if \(E \subset {\bf R}^2\) is measurable and bounded and if \( {\overline \partial}f(x) = 0 \) a.e. in E, then \( \int_E \vert D f(x) \vert^{p} dx < \infty \hskip10pt {\rm for} \hskip6pt p = \frac{2K}{K-1}. \)