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 {{mathrm{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 Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), 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 found so that if uin W^{1,q_{sigma _1,sigma _2}}(Omega ) is solution to the elliptic equation then nabla uin L^{p_{sigma _1,sigma _2}}_{mathrm{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 Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.