Title: 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.