Inserted: 20 dec 2016
Last Updated: 22 sep 2017
Journal: Calc. Var. and PDE
Let $\Omega$ be an open subset of a Stein manifold $\Sigma$ and let $M$ be its boundary. It is well knwon that $M$ inherits a natural contact structure. In this paper we consider a family of variational functionals $F_\varepsilon$ defined by the sum of two terms: a Dirichlet-type energy associated with a subriemannian structure in $\Omega$ and a potential term on the boundary $M$. We prove that the functionals $F_\varepsilon$ $\Gamma$-converge to the intrinsic perimeter in $M$ associated with its contact structure.
Similar results have been obtained in the Euclidean space by Alberti, Bouchitté, Seppecher. We stress that already in the Euclidean setting the situation is not covered by the classical Modica-Mortola Theorem because of the precense of the boundary term.
We recall also that Modica-Mortola type results (without a boundary term) have been proved in the Euclidean space for subriemannian energies by Monti and Serra Cassano.