| Abstract: |
| We investigate the fractional $s$-perimeter relative to a bounded domain $\Omega$ in the Grushin space, a typical model of sub-Riemannian manifolds that generally lack a group structure. On the one hand, we establish a limiting formula for the $s$-perimeter as $s\rightarrow0^+$, imposing no regularity assumptions on the boundary of $\Omega$. On the other hand, we provide a converse under a mild boundary condition that allows for domains with highly irregular (e.g., fractal) boundaries. Our results significantly weakens the typical Lipschitz or $C^{1,\alpha}$ regularity required in existing literature. The proofs exploit intrinsic properties of the Grushin heat semigroup, thereby circumventing tools such as the $L^\infty$-Liouville property used in earlier Riemannian settings. |
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