Harmonic measure on sets of codimension larger than one Article Swipe

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Guy David , Joseph Feneuil , Svitlana Mayboroda ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.1016/j.crma.2017.02.013 · OA: W2491208050

We introduce a new notion of a harmonic measure for a d -dimensional set in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>d</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math>, that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the d -dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.