Search results for "Isoperimetric dimension"

Isoperimetric inequality via Lipschitz regularity of Cheeger-harmonic functions

2014

Abstract Let ( X , d , μ ) be a complete, locally doubling metric measure space that supports a local weak L 2 -Poincare inequality. We show that optimal gradient estimates for Cheeger-harmonic functions imply local isoperimetric inequalities.

The isoperimetric inequality and the geodesic spheres. Some geometric consequences

1986

Isoperimetric inequality from the poisson equation via curvature

2012

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-Poincare 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.

Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds

2012

Abstract We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumptions, this can happen only on topological spheres. To cite this article: R. Grimaldi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Fl�chen Beschr�nkter Mittlerer Kr�mmung in Einer Dreidimensionalen Riemannschen Mannigfaltigkeit

1973

In recent papers HILDEBRANDT [11] and HARTH [5] proved the existence of solutions of the problem of Plateau for surfaces of bounded mean curvature with fixed and free boundaries in E3 and for minimal surfaces with free boundaries in a Riemannian manifold, respectively. Here their methods will be combined to solve the problem of Plateau for surfaces of bounded mean curvature in a Riemannian manifold. This will be done for fixed and free boundaries. Moreover, isoperimetric inequalities for the solutions will be given.

Remark on a nonlocal isoperimetric problem

2017

Abstract We consider isoperimetric problem with a nonlocal repulsive term given by the Newtonian potential. We prove that regular critical sets of the functional are analytic. This optimal regularity holds also for critical sets of the Ohta–Kawasaki functional. We also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set.

Bonnesenʼs inequality for John domains in Rn

2012

Abstract We prove sharp quantitative isoperimetric inequalities for John domains in R n . We show that the Bonnesen-style inequalities hold true in R n under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7] and Fuglede (1989) [4] to reduce our problem to symmetric domains.