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 essentially proportional to the Hausdorff measure on the boundary. Purely unrectifiable boundaries of domains are often characterised by the harmonic measure being singular with respect to the Hausdorff measure on the boundary. Our result provides more examples of purely unrectifiable boundaries for which the analogous results do not hold for general elliptic operators. 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.