Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds
By Mattia Fogagnolo, L. Mazzieri. Journal of Functional Analysis, 2020
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The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull $\Omega^*$ of a set $\Omega$ is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where $\Omega$ has $\mathscr{C}^{1, \alpha}$-boundary, the area of $\partial \Omega^*$ is recovered as the limit of the $p$-capacities of $\Omega$, as $p \to 1^+$. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided $3 \leq n \leq 7$.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/1a47cb842c51d95fce4a7ff3f2a713c21a423c08
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In Applied Mathematics
Citations in the corpus
- Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “The smoothness of the boundary of the isoperimetric set is an essential requirement not only in our argument, but also in Agostiniani, Fogagnolo and Mazzieri [1, 25] and Brendle [7].”