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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