Oxide heterostructures are of great interest both for fundamental and applicative reasons. In particular the two-dimensional electron gas at the LaAlO$_3$/SrTiO$_3$ or LaTiO$_3$/SrTiO$_3$ interfaces displays many different physical properties and functionalities. However there are clear indications that the interface electronic state is strongly inhomogeneous and therefore it is crucially relevant to investigate possible intrinsic electronic mechanisms underlying this inhomogeneity. Here the electrostatic potential confining the electron gas at the interface is calculated self-consistently, finding that the electron confinement at the interface may induce phase separation, to avoid a thermodynamically unstable state with a negative compressibility. This provides a generic robust and intrinsic mechanism for the experimentally observed inhomogeneous character of these interfaces.
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