*Preprint*

**Inserted:** 28 dec 2011

**Last Updated:** 25 jan 2012

**Year:** 2011

**Abstract:**

This paper deals with higher gradient integrability for $\sigma$-harmonic functions $u$ with discontinuous coefficients $\sigma$, i.e. weak solutions of $\mbox{div }(\sigma \nabla u) = 0$.
We focus on two-phase conductivities $\sigma:\Omega\subset\mathbb R^2\mapsto \{\sigma_1,\sigma_2\}\subset M^{2\times 2}$, and study the higher integrability of the corresponding gradient field $

\nabla u

$.
The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement described by the sets $E_i=\sigma^{-1}(\sigma_i)$. We find the optimal integrability exponent of the gradient field corresponding to any pair $\{\sigma_1,\sigma_2\}$ of positive definite matrices, i.e., the worst among all possible microgeometries. We also show that it is attained by so-called exact solutions of the corresponding PDE.
Furthermore, among all two-phase conductivities with fixed ellipticity, we characterize those that correspond to the worse integrability.

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