Counterexamples to the typical behaviour of elliptic measures on fractals

par Polina Perstneva (Paris Saclay)

lundi 25 mars 2024, 14:00 → 15:00 Europe/Paris

Amphi Schwartz

Recent developments in Geometric Measure Theory have led to the understanding that, essentially, rectifiability of the boundary of a domain is necessary and sufficient for the harmonic measure to be (qualitatively) absolutely continuous with respect to the Hausdorff measure on that boundary. The counterpart of this is the following: for purely unrectifiable sets, the harmonic measure is singular with respect to the boundary measure. Recall that a pure unrectifiable set is an anti-pod of a rectifiable set: its intersection with any image of a Lipschitz function is almost empty. Classical examples of unrectifiable sets are fractals: on the plane, the famous ones are the Koch snowflake, the four corners Cantor set, the Sierpinski carpet, etc. 

It is also known that all the operators close to the Laplacian, which generates the harmonic measure, produce elliptic measures with the same properties as above. However, it turns out that for some unrectifiable sets on the plane there exists an elliptic operator with a scalar coefficient whose elliptic measure behaves in the opposite way than the harmonic measure does. We will discuss these counterexamples discovered in the last couple of years by G. David, S. Mayboroda, and the speaker, and look briefly into some open problems around them.