Isoperimetric inequality from the poisson equation via curvature
Article first published online: 11 MAY 2012
Copyright © 2012 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 65, Issue 8, pages 1145–1168, August 2012
How to Cite
Jiang, R. and Koskela, P. (2012), Isoperimetric inequality from the poisson equation via curvature. Comm. Pure Appl. Math., 65: 1145–1168. doi: 10.1002/cpa.21405
- Issue published online: 22 MAY 2012
- Article first published online: 11 MAY 2012
- Manuscript Revised: OCT 2011
- Manuscript Received: FEB 2011
In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q-regular measure, where Q > 1, that supports a local L2-Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L∞-norm of the gradient Du can be bounded by the Lorentz norm LQ,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on . © 2011 Wiley Periodicals, Inc.