It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely unrectifiable) Cantor sets in R2 whose elliptic measure is absolutely continuous, and in fact, essentially proportional to the Hausdorff measure.
Bibliographical noteFunding Information:
David was partially supported by the European Community H2020 grant GHAIA 777822 , and the Simons Foundation grant 601941 , GD. Mayboroda was supported in part by the Alfred P. Sloan Fellowship , the NSF grants DMS 1344235 , DMS 1839077 , and The Simons Foundation grant 563916 , SM.
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- Cantor set
- Elliptic operator
- Green function
- Harmonic measure