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(s?u) = 0 in dimension two, in the case when the essential range of s consists of only two elliptic matrices, i.e., s ? {s1,s2} a.e. in O. In [9], for every pair of elliptic matrices s1 and s2 exponents ps1,s2 ? (2,+8) and qs1,s2 ? (1,2) have been found so that if u ? W1,qs1,s2(O) is solution to the elliptic equation then ?u ? Lps1,s2(O) and the optimality of the upper exponent ps1,s2 has been proved. In this paper we complement the above result by proving the optimality of the lower exponent qs1,s2. Precisely, we show that for every arbitrarily small d, one can find a particular microgeometry, i.e. an arrangement of the sets s-1(s1) and s-1(s2), for which there exists a solution u to the corresponding elliptic equation such that ?u ? Lqs1,s2-d, but ?u ? Lqs1,s2-d. 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.