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Isoperimetric inequality from the Poisson equation via curvature. (English) Zbl 1247.53045

It is well known that the isoperimetric inequality in ${ℝ}^{n}$ is a consequence of the Sobolev embedding ${W}_{0}^{1,1}\to {L}^{\frac{n}{n-1}}$. Another way to derive the isoperimetric inequality in ${ℝ}^{n}$ is to look at solutions of the Poisson equation ${\Delta }u=g$ and to use the inequality ${\parallel Du\parallel }_{{L}^{\infty }\left(B\right)}\le c{\parallel g\parallel }_{{L}^{n,1}\left(2B\right)}$. (This argument, which does not seem to appear in the literature, is given in the introduction of the paper under review.)

In the paper under review an isoperimetric inequality for metric measure spaces satisfying a local ${L}^{2}$-Poincaré inequality is proved under suitable assumptions. The setting is that of a complete, path-connected metric space $\left(X,d\right)$ with a $Q$-regular measure $\mu$ (which means roughly that volume of balls grows like ${R}^{Q}$) and which satisfies a quantitative local ${L}^{2}$-Poincaré-inequality. The Sobolev space ${H}^{1,2}\left(X\right)$ is the completion of the space of locally Lipschitz functions for the ${H}^{1,2}$-norm. For functions $u\in {H}_{loc}^{1,2}\left(x\right)$ one has the Cheeger derivative $Du$ and one can use this to define what a solution of the Cheeger-Poisson equation ${\Delta }u=g$ is. The assumption made in the paper under review is that solutions to the Cheeger-Poisson equation satisfy the inequality

${\parallel |Du|\parallel }_{{L}^{\infty }\left(B\right)}\le C{\left({R}^{-1}{\int }_{2B}{|u|}^{2}d\mu \right)}^{\frac{1}{2}}+{\parallel g\parallel }_{{L}^{Q,1}\left(2B\right)}$

for $B=B\left(x,r\right)$ with $2r.

Under these assumptions the authors prove that a local ${L}^{2}$-Poincaré inequality implies an isoperimetric inequality

$\mu {\left(E\right)}^{\frac{Q-1}{Q}}\le CP\left(E,X\right)$

for bounded Borel sets $E\subset X$, where $P\left(E,X\right)$ denotes the perimeter of $E$ in $X$. As a corollary they obtain the local Sobolev inequality

${\parallel \phi \parallel }_{{L}^{\frac{Q}{Q-1}}\left(X\right)}\le {C\parallel |D\phi |\parallel }_{{L}^{1}\left(X\right)}$

for Lipschitz functions $\phi$ supported in a ball of given radius.

The assumptions made in the paper under review are in particular satisfied for Riemannian manifolds of nonnegative Ricci curvature, maximal volume growth and dimension $\ge 3$. There is a generalized notion of lower Ricci curvature bounds for metric measure spaces due to Lott-Villani and Sturm. It follows from results of Lott-Villani, von Renesse and Rajala that this curvature bound implies local ${L}^{1}$-Poincaré-inequalities, hence the results of the paper under review can be applied in this setting.

##### MSC:
 53C23 Global geometric and topological methods; differential geometric analysis on metric spaces 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global) 54E35 Metric spaces, metrizability 49J52 Nonsmooth analysis (other weak concepts of optimality) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems