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How to Find a Measure from its Potential

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Abstract

We consider the problem of finding a measure from the given values of its logarithmic potential on the support. It is well known that a solution to this problem is given by the generalized Laplacian. The case of our main interest is when the support is contained in a rectifiable curve, and the measure is absolutely continuous with respect to the arclength on this curve. Then the generalized Laplacian is expressed by a sum of normal derivatives of the potential. Such a representation is already available for smooth curves, and we show it holds for any rectifiable curve in the plane. We also relax the assumptions imposed on the potential.

Finding a measure from its potential often leads to another closely related problem of solving a singular integral equation with Cauchy kernel. The theory of such equations is well developed for smooth curves. We generalize this theory to the class of Ahlfors regular curves and arcs, and characterize the bounded solutions on arcs.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Igor Pritsker

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  1. Igor Pritsker
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Correspondence to Igor Pritsker.

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Dedicated to Professor W. K. Hayman on his 80th birthday

Research was partially supported by the National Security Agency under grant H98230-06-1-0055, and by the Alexander von Humboldt Foundation.

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Pritsker, I. How to Find a Measure from its Potential. Comput. Methods Funct. Theory 8, 597–614 (2008). https://doi.org/10.1007/BF03321707

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  • DOI: https://doi.org/10.1007/BF03321707

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