Existence of quadrature surfaces for positive measures with finite support

Abstract

Let μ be a positive measure with finite support in ℜn (n≧2). Then we show that there is a bounded open set Ω, containing the support of μ, whose single-layer potential coincides with the potential of μ outside Ω.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Alt, H. W. and Caffarelli, L. A.: ‘Existence and Regularity for a Minimum Problem with Free Boundary’,J. Reine Angew. Math. 325 (1981), 434–448.

    Google Scholar 

  2. 2.

    Shahgholian, H.: ‘Quadrature Surfaces as Free Boundaries’, preprint.

  3. 3.

    Federer, H.:Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1969).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Shahgholian, H. Existence of quadrature surfaces for positive measures with finite support. Potential Anal 3, 245–255 (1994). https://doi.org/10.1007/BF01053435

Download citation

Mathematics Subject Classifications (1991)

  • 31B20
  • 35J05

Key words

  • Quadrature surface
  • positive Dirac masses
  • harmonic functions