Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

Abstract

We study the higher gradient integrability of distributional solutions $u$ to the equation $div(\sigma \nabla u) = 0$ in dimension two, in the case when the essential range of $\sigma$ consists of only two elliptic matrices, i.e., $\sigma\in\{\sigma_1, \sigma_2\}$ a.e. in $\Omega$. In [4], for every pair of elliptic matrices $\sigma_1$ and $\sigma_2$, exponents $p_{\sigma_1,\sigma_2}\in(2,+\infty)$ and $q_{\sigma_1,\sigma_2}\in (1,2)$ have been characterised so that if $u\in W^{1,q_{\sigma_1,\sigma_2}}(\Omega)$ is solution to the elliptic equation then $\nabla u\in L^{p_{\sigma_1,\sigma_2}}_{\rm weak}(\Omega)$ and the optimality of the upper exponent $p_{\sigma_1,\sigma_2}$ has been proved. In this paper we complement the above result by proving the optimality of the lower exponent $q_{\sigma_1,\sigma_2}$. Precisely, we show that for every arbitrarily small $\delta$, one can find a particular microgeometry, i.e., an arrangement of the sets $\sigma^{-1}(\sigma_1)$ and $\sigma^{-1}(\sigma_2)$, for which there exists a solution $u$ to the corresponding elliptic equation such that $\nabla u \in L^{q_{\sigma_1,\sigma_2}-\delta}$, but $\nabla u \notin L^{q_{\sigma_1,\sigma_2}}.$ The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in [2] for the isotropic case.