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

الأصل · EN

We obtain an estimate for the Hölder 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 ² and, for every x∈Ω, A(x) is a 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 A(x)≡1. Furthermore, we show that our estimate is sharp. Indeed, for every τ∈[0,1] we construct coefficient matrices Aτ such that A₀ is isotropic and A₁ has unit determinant, and such that our estimate for Aτ reduces to an equality, for every τ∈[0,1].