On the best Holder exponent for two dimensional elliptic equations in divergence form

We obtain an estimate for the Holder continuity exponent for weak solutions to the following elliptic equation in divergence form: div (A(x)del u) = 0 in Omega, where. is a bounded open subset of R-2 and, for every x epsilon Omega, A(x) is a symmetric matrix with bounded measurable coefficients. Such an estimate "interpolates" between the well-known estimate of Piccinini and Spagnolo in the isotropic case A(x) = a(x) I, where a is a bounded measurable function, and our previous result in the unit determinant case det A equivalent to 1. Furthermore, we show that our estimate is sharp. Indeed, for every tau epsilon [0, 1] we construct coefficient matrices At such that A(0) is isotropic and A(1) has unit determinant, and such that our estimate for At reduces to an equality, for every tau epsilon [0, 1].