Uploaded August 2026 | Updated September 2026, 3 weeks ago
Quantitative Rectifiability and Harmonic Measure
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A set in the Euclidean space is called n-rectifiable if it is almost all contained in a countable union of C1 n-dimensional manifolds. The theory of quantitative rectifiability studies this property using tools from harmonic analysis, such as square functions and singular integrals.On the other hand, harmonic measure is a fundamental notion in the solution of the Dirichlet problem for the Laplace equation and has important applications in complex analysis. For a bounded domain, the harmonic measure of a subset of the boundary coincides with the probability that a Brownian motion starting inside the domain exits the domain through that subset. An important and old problem in analysis consists in understanding the relationship between harmonic measure and surface measure in a given domain. The notion of rectifiability plays a central role in this problem. In this talk we will survey classical results and recent advances on this topic obtained using tools from quantititative rectifiability. In particular, we will describe the solution of the one-phase and two-phase problems for harmonic measure and new results about the solvability of the Dirichlet and regularity problems in rough domain in Lp.
Quantitative Rectifiability and Harmonic Measure
~
A set in the Euclidean space is called n-rectifiable if it is almost all contained in a countable union of C1 n-dimensional manifolds. The theory of quantitative rectifiability studies this property using tools from harmonic analysis, such as square functions and singular integrals.On the other hand, harmonic measure is a fundamental notion in the solution of the Dirichlet problem for the Laplace equation and has important applications in complex analysis. For a bounded domain, the harmonic measure of a subset of the boundary coincides with the probability that a Brownian motion starting inside the domain exits the domain through that subset. An important and old problem in analysis consists in understanding the relationship between harmonic measure and surface measure in a given domain. The notion of rectifiability plays a central role in this problem. In this talk we will survey classical results and recent advances on this topic obtained using tools from quantititative rectifiability. In particular, we will describe the solution of the one-phase and two-phase problems for harmonic measure and new results about the solvability of the Dirichlet and regularity problems in rough domain in Lp.
