A Two Phase Free Boundary Problem Related to Quadrature Domains |
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| Authors: | Behrouz Emamizadeh Jyotshana V Prajapat Henrik Shahgholian |
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| Institution: | 1.Department of Mathematics,The Petroleum Institute,Abu Dhabi,UAE;2.Department of Mathematics,Royal Institute of Technology,Stockholm,Sweden |
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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
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$ - \Delta u = (\mu_+ - \lambda_+ )\chi_{\{u > 0\}} - (\mu_- - \lambda_- )\chi_{\{u < 0\}} ~~~{\rm in }~~~ {I\!\!R}^N. $ - \Delta u = (\mu_+ - \lambda_+ )\chi_{\{u > 0\}} - (\mu_- - \lambda_- )\chi_{\{u < 0\}} ~~~{\rm in }~~~ {I\!\!R}^N. |
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