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

Abstract:

We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form: \[ \mathrm {div}(A(x)\nabla u)=0 \qquad \mathrm {in} \Omega , \] where $\Omega$ is a bounded open subset of $\mathbb R^2$ and, for every $x\in \Omega$, $A(x)$ 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 $\det A\equiv 1$. Furthermore, we show that our estimate is sharp. Indeed, for every $\tau \in [0,1]$ we construct coefficient matrices $A_\tau$ such that $A_0$ is isotropic and $A_1$ has unit determinant, and such that our estimate for $A_\tau$ reduces to an equality, for every $\tau \in [0,1]$.