Dahlberg's theorem in higher co-dimension |
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Authors: | Guy David Joseph Feneuil Svitlana Mayboroda |
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Institution: | 1. Univ Paris-Sud, Laboratoire de Mathématiques, UMR8628, Orsay, F-91405, France;2. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA |
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Abstract: | In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension in , and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in , , with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of and the Hausdorff measure. |
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Keywords: | 42B37 31B25 31B25 35J25 35J70 Boundary with co-dimension higher than 1 Degenerate elliptic operators Dahlberg's theorem Harmonic measure in higher codimension |
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