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| In this talk I will present a $\Gamma$-convergence type result for a boundary phase transition model in an almost K\"ahler manifold $M$. The functional that we consider involves a Dirichlet energy in the interior of the manifold and a double-well potential term on its boundary. Locally, and close to the boundary, the manifold $M$ is diffeomorphic to $\mathbb H^n\times \mathbb R$, and therefore the model case for our functional is of the form
$$\int_{\mathbb H^n\times \mathbb R}( |\nabla_{\mathbb H} u(\xi, z)|^2 +|\partial_z u|^2) d\xi dz + \int_{\mathbb H ^n} W(u) d\xi,$$
where $(\xi,z)\in \mathbb H^n \times \mathbb R$ and $\nabla_{\mathbb H}$ denotes the horizontal gradient in $\mathbb H^n$. In our main result we establish that, after a suitable rescaling, this energy functional $\Gamma$-converges to the (intrinsic) perimeter functional on the boundary of $M$.
This is the analogue, for almost K\"ahler manifolds, of a (Euclidean) $\Gamma$-convergence type result by Alberti, Bouchitt\'e and Seppecher. |
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