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We derive various sharp upper bounds for the $p$-capacity of a smooth compact set $K$ in the hyperbolic space $\mathbb{H}^n$ and the Euclidean space $\mathbb{R}^n$. Firstly, using the inverse mean curvature flow, for the mean convex and star-shaped set $K$ in $\mathbb{H}^n$, we obtain sharp upper bounds for the $p$-capacity $\mathrm{Cap}_p(K)$ in three cases: (1) $n\geq 2$ and $p=2$, (2) $n=2$ and $p\geq 3$, (3) $n=3$ and $11$. Secondly, for the compact set $K$ in $\mathbb{R}^3$, using the weak inverse mean curvature flow, we get a sharp upper bound for the $p$-capacity ($1Comment: 23 pages; all comments are welcome. arXiv admin note: text overlap with arXiv:2104.09905 |
