$\Gamma$-convergence of variational functionals with boundary terms in Stein manifolds

TL;DR

This paper proves that a family of variational functionals with boundary terms on Stein manifolds converges to the intrinsic perimeter, extending Euclidean results to complex geometric settings with boundary considerations.

Contribution

It establishes $ ext{Gamma}$-convergence of boundary-including variational functionals in Stein manifolds, generalizing known Euclidean results to complex geometric contexts.

Findings

Functional $F_ ext{varepsilon}$ $ ext{Gamma}$-converges to perimeter in contact boundary.

Extends Euclidean $ ext{Gamma}$-convergence results to Stein manifolds.

Addresses boundary term challenges not covered by classical theorems.

Abstract

Let $\Omega$ be an open subset of a Stein manifold $\Sigma$ and let $M$ be its boundary. It is well known 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 sub-Riemannian 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, Bouchitt\'e, Seppecher. We stress that already in the Euclidean setting the situation is not covered by the classical Modica-Mortola Theorem because of the presence of the boundary term. We recall also that Modica-Mortola type results (without a boundary term) have been proved in the Euclidean space for sub-Riemannian energies by Monti and Serra Cassano.