zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Isoperimetric inequality from the Poisson equation via curvature. (English) Zbl 1247.53045

It is well known that the isoperimetric inequality in n is a consequence of the Sobolev embedding W 0 1,1 L n n-1 . Another way to derive the isoperimetric inequality in n is to look at solutions of the Poisson equation Δu=g and to use the inequality Du L B cg L n,1 2B . (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 X,d with a Q-regular measure μ (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 X is the completion of the space of locally Lipschitz functions for the H 1,2 -norm. For functions uH loc 1,2 x one has the Cheeger derivative Du and one can use this to define what a solution of the Cheeger-Poisson equation Δu=g is. The assumption made in the paper under review is that solutions to the Cheeger-Poisson equation satisfy the inequality

|Du| L B CR -1 2B |u| 2 dμ 1 2 +g L Q,1 2B

for B=Bx,r with 2r<R.

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

μE Q-1 Q CPE,X

for bounded Borel sets EX, where PE,X denotes the perimeter of E in X. As a corollary they obtain the local Sobolev inequality

ϕ L Q Q-1 X C|Dϕ| L 1 X

for Lipschitz functions ϕ 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 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:
53C23Global geometric and topological methods; differential geometric analysis on metric spaces
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
54E35Metric spaces, metrizability
49J52Nonsmooth analysis (other weak concepts of optimality)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems