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

Abstract

We obtain an estimate for the Holder continuity exponent for weak solutions to the following elliptic equation in divergence form: div(A(x)∇u) =0 in Ω, where Ω is a bounded open subset of R 2 and, for every x ∈ Ω, 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 ≡ 1. Furthermore, we show that our estimate is sharp. Indeed, for every r ∈ [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 r ∈ [0,1].