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

We study the higher gradient integrability of distributional solutions $u$ to the equation $div(\sigma \mathrm{\nabla }u)=0$ 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 $\mathrm{\Omega }$. In [4], for every pair of elliptic matrices ${\sigma }_1$ and ${\sigma }_2$, exponents $p_{{\sigma }_1,{\sigma }_2}\in (2,+\mathrm{\infty })$ and $q_{\sigma_1,\sigma_2}\in (1,2)$havebeencharacterisedsothatif$u\in W^{1,q_{\sigma_1,\sigma_2}}(\Omega)$issolutiontotheellipticequationthen$\nabla u\in L^{p_{\sigma_1,\sigma_2}}{\rm weak}(\Omega)$ 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.