Search results for "Minkowski inequality"

The Poincaré inequality is an open ended condition

2008

Let p > 1 and let (X,d,µ) be a complete metric measure space with µ Borel and doubling that admits a (1,p)-Poincare inequality. Then there exists e > 0 such that (X,d,µ) admits a (1,q)-Poincare inequality for every q > p - e, quantitatively.

On the Euler-Lagrange inequality of a convex variational integral in Orlicz spaces

1987

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.

On improved fractional Sobolev–Poincaré inequalities

2016

We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

An overdetermined problem for the anisotropic capacity

2015

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in \({\mathbb {R}}^{N}\), establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm \(H_0\)).

Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality

2016

We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.