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We consider an energy functional combining the square of the localoscillation of a one--dimensional function with a double well potential. Weestablish the existence of minimal heteroclinic solutions connecting the twowells of the potential.  This existence result cannot be accomplished by standard methods, due to thelack of compactness properties.  In addition, we investigate the main properties of these heteroclinicconnections. We show that these minimizers are monotone, and therefore theysatisfy a suitable Euler-Lagrange equation.  We also prove that, differently from the classical cases arising in ordinarydifferential equations, in this context the heteroclinic connections are notnecessarily smooth, and not even continuous (in fact, they can be piecewiseconstant). Also, we show that heteroclinics are not necessarily unique up to atranslation, which is also in contrast with the classical setting.  Furthermore, we investigate the associated Dirichlet problem, studyingexistence, uniqueness and partial regularity properties, providing explicitsolutions in terms of the external data and of the forcing source, andexhibiting an example of discontinuous solution.

Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type