Equilibrium measures on curves

Speaker(s)

Damian Dąbrowski

Affiliation

IMPAN

Language of the talk

English

Date

Oct. 8, 2025, 12:30 p.m.

Room

room 4060

Seminar

Seminarium Zakładu Równań i Analizy

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In this talk I will discuss one of the most classical objects of study in potential theory: equilibrium measures for the logarithmic energy. Given a compact set E, the (logarithmic) equilibrium measure on E is the unique (if it exists) minimizer of the logarithmic energy among all measures supported on E. In the case of planar sets, the equilibrium measure coincides with the harmonic measure, and is rather well-understood. However, almost nothing is known about the equilibrium measures associated to subsets of higher dimensional Euclidean spaces. In a recent paper with Tuomas Orponen we show that these measures are absolutely continuous with respect to the arc-length measure on C^{1,alpha} curves in arbitrary dimension. I will describe some ideas of our proof, and mention many related open problems.

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