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A Two Phase Free Boundary Problem Related to Quadrature Domains

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Abstract

In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution to

$$ - \Delta u = (\mu_+ - \lambda_+ )\chi_{\{u > 0\}} - (\mu_- - \lambda_- )\chi_{\{u < 0\}} ~~~{\rm in }~~~ {I\!\!R}^N. $$

where \(\lambda_\pm >0 \) are given constants and \(\mu^\pm\), are non-negative bounded Radon measures, with compact support. Our primary concern is to discuss existence and geometric properties of solutions.

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Authors and Affiliations

  1. Department of Mathematics, The Petroleum Institute, P.O. 2533, Abu Dhabi, UAE

    Behrouz Emamizadeh & Jyotshana V. Prajapat

  2. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Henrik Shahgholian

Authors
  1. Behrouz Emamizadeh
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  2. Jyotshana V. Prajapat
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  3. Henrik Shahgholian
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Corresponding author

Correspondence to Henrik Shahgholian.

Additional information

H. Shahgholian was supported in part by Swedish Research Council. This work was started when H. Shahgholian was visiting The Petroleum Institute. The authors are grateful to the referee for valuable suggestions and comments.

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Emamizadeh, B., Prajapat, J.V. & Shahgholian, H. A Two Phase Free Boundary Problem Related to Quadrature Domains. Potential Anal 34, 119–138 (2011). https://doi.org/10.1007/s11118-010-9184-y

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  • DOI: https://doi.org/10.1007/s11118-010-9184-y

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