Gradient integrability and rigidity results for two-phase conductivities in two dimensions

Abstract

This paper deals with higher gradient integrability for $\sigma$-harmonic functions $u$ with discontinuous coefficients $\sigma$, i.e. weak solutions of $\mathrm{div}(\sigma \nabla u)=0$ in dimension two. When $\sigma$ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and $\sigma$ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries.

We focus also on two-phase conductivities, i.e., conductivities assuming only two matrix values, ${\sigma }_1$ and ${\sigma }_2$, and study the higher integrability of the corresponding gradient field $|\nabla u|$ for this special but very significant class. 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 elliptic matrices, i.e., the worst among all possible microgeometries.

We also treat the unconstrained case when an arbitrary but finite number of phases are present.