| Résume | 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 absolutely continuous with respect to the Hausdorff measure on that boundary. Consequently, for purely unrectifiable sets (e.g., the Koch snowflake, the four corners Cantor set, the Sierpinski carpet, etc), the harmonic measure is singular with respect to the boundary measure. It is also known that all operators close to the Laplacian, which generates the harmonic measure, produce elliptic measures with the same properties as above. However, it has been discovered lately that for some unrectifiable sets on the plane, there exists an elliptic operator with a scalar coefficient whose elliptic measure is proportional to the boundary measure. We will discuss the proof scheme of these results and some perspectives on their extension to higher dimensions or wider classes of unrectifiable sets. |
