  In this paper we introduce the multi-phase version of the so-called
Quadrature Domains (QD), which refers to a generalized type of mean value
property for harmonic functions. The well-established and developed theory of
one-phase QD was recently generalized to a two-phase version, by one of the
current authors (in collaboration). Here we introduce the concept of the
multi-phase version of the problem, and prove existence as well as several
properties of such solutions. In particular, we discuss possibilities of
multi-junction points.
