  We develop a theory of the domain patterns in systems with competing
short-range attractive interactions and long range repulsive Coulomb
interactions. We take an energetic approach, in which patterns are considered
as critical points of a mean-field free energy functional. Close to the
microphase separation transition, this functional takes on a universal form,
allowing to treat a number of diverse physical situations within a unified
framework. We use asymptotic analysis to study domain patterns with sharp
interfaces. We derived an interfacial representation of the pattern's free
energy which remains valid in the fluctuating system, with a suitable
renormalization of the Coulomb interaction's coupling constant. We also derived
integrodifferential equations describing the stationary domain patterns of
arbitrary shapes and their thermodynamic stability, coming from the first and
second variation of the interfacial free energy. We showed that the length
scale of a stable domain pattern must obey a certain scaling law with the
strength of the Coulomb interaction. We analyzed existence and stability of
localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns
in two dimensions. We showed that these patterns are metastable in certain
ranges of the parameters and that they can undergo morphological instabilities
leading to the formation of more complex patterns. We discuss nucleation of the
domain patterns by thermal fluctuations and pattern formation scenarios for
various thermal quenches. We argue that self-induced disorder is an intrinsic
property of the domain patterns in the systems under consideration.
