Good elliptic operators on snowflakes

Abstract

We construct elliptic operators with scalar coefficients on the complements $(\mathbb{R}^2 \setminus S)^+$ of some Koch-type snowflakes $S$, whose Hausdorff dimensions cover the full range $(1, \ln{(4)}/\ln{(3)})$, such that the operator's elliptic measures are equal to the Hausdorff measure on the boundary. This provides another example of the phenomenon that, though purely unrectifiable boundaries of domains are often characterised by the harmonic measure being singular with respect to the Hausdorff measure on the boundary, for some purely unrectifiable boundaries one can construct an elliptic operator whose elliptic measure behaves in a drastically different way. Plus, in $\mathbb{R}^2$, this operator can be chosen in a way that its coefficient is scalar, as opposed to a $2 \times 2$ matrix-valued one.