Séminaire du pôle analyse 3/12 : Polina Perstneva (CMAP)
03 Dec. 2024 - 03 Dec. 2024
Title: Elliptic measure in domains with boundaries of codimension different from 1
Abstract: The elliptic measure allows one to reconstruct the solution of the standard Dirichlet problem with an elliptic operator from the continuous boundary data. However, the solvability of the Dirichlet problem with rough boundary data is connected to the absolute continuity of the elliptic measure with respect to the boundary measure on the domain.
Recent developments have led to the understanding that, essentially, $n - 1$-rectifiability of the boundary of a domain in $\R^n$ is necessary and sufficient for the harmonic measure, a special case of elliptic measure for the Laplacian operator, to be absolutely continuous with respect to the Hausdorff measure on that boundary. It is also known that all the operators close to the Laplacian produce elliptic measures with the same properties. However, it turns out that there are elliptic measures which behave very differently from the class of measures mentioned above. In the first part of the talk, we will discuss these counterexamples discovered in the last couple of years.
The motivation for these results came from a recent attempt to characterise $d$-rectifiability of sets in $\R^n$ with $d < n - 1$ in terms of analytic properties of some analogue of the harmonic measure. We will discuss this in the second part of the talk.
Most of the talk will be accessible to people unfamiliar with the field.
Recent developments have led to the understanding that, essentially, $n - 1$-rectifiability of the boundary of a domain in $\R^n$ is necessary and sufficient for the harmonic measure, a special case of elliptic measure for the Laplacian operator, to be absolutely continuous with respect to the Hausdorff measure on that boundary. It is also known that all the operators close to the Laplacian produce elliptic measures with the same properties. However, it turns out that there are elliptic measures which behave very differently from the class of measures mentioned above. In the first part of the talk, we will discuss these counterexamples discovered in the last couple of years.
The motivation for these results came from a recent attempt to characterise $d$-rectifiability of sets in $\R^n$ with $d < n - 1$ in terms of analytic properties of some analogue of the harmonic measure. We will discuss this in the second part of the talk.
Most of the talk will be accessible to people unfamiliar with the field.