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 (σ∇ u) = 0 in dimension two, in the case when the essential range of σ consists of only two elliptic matrices, i.e., σ∈ σ1, σ2 a.e. in Ω. In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices σ1 and σ2, exponents pσ1,2,+∞) and qσ1,1,2) have been found so that if u∈W1,qσ1,σ2(Ω) is solution to the elliptic equation then ∇u∈Lweakpσ1,σ2(Ω) and the optimality of the upper exponent pσ1,σ2 has been proved. In this paper we complement the above result by proving the optimality of the lower exponent qσ1,σ2. Precisely, we show that for every arbitrarily small δ, one can find a particular microgeometry, i.e., an arrangement of the sets σ- 1(σ1) and σ- 1(σ2) , for which there exists a solution u to the corresponding elliptic equation such that ∇u∈Lqσ1,σ2-δ, but ∇u∉Lqσ1,σ2. The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.