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Extremal functions for modules of systems of measures

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Abstract

We extend a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in R2, to Rn making use of Fuglede’s p-module of systems of measures. The extremal functions are identified and the p-module of systems of measures is computed in condensers of rather general type and in their images under homeomorphisms of certain regularity. At the beginning, we discuss and apply Rodin’s Theorem in order to obtain estimates for the conformal modules of parallelograms and ring domains in terms of directional dilatations.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Fordham University, Bronx, NY, 10458-5165, USA

    Melkana Brakalova

  2. Department of Mathematics, University of Bergen, P.O. Box 7800, Bergen, N-5020, Norway

    Irina Markina

Authors
  1. Melkana Brakalova
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  2. Irina Markina
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  3. Alexander Vasil’ev
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Corresponding author

Correspondence to Melkana Brakalova.

Additional information

The first author was partially supported by a Faculty Research Grant, Fordham University, USA.

The second and third authors were partially supported by the grants of the Norwegian Research Council #239033/F20 and #213440/BG and by EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669Z

Alexander Vasil’ev passed away on October 19, 2016.

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Brakalova, M., Markina, I. & Vasil’ev, A. Extremal functions for modules of systems of measures. JAMA 133, 335–359 (2017). https://doi.org/10.1007/s11854-017-0036-1

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  • DOI: https://doi.org/10.1007/s11854-017-0036-1

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