Isoperimetric inequality for curves with curvature bounded below

For embedded closed curves with curvature bounded below, we prove an isoperimetric inequality estimating the minimal area bounded by such curves for a fixed perimeter.     

Isoperimetric inequality via Lipschitz regularity of Cheeger-harmonic functions

Let (X,d,μ)$(X,d,\mu )$ be a complete, locally doubling metric measure space that supports a local weak L²$L^2$-Poincaré inequality. We show that optimal gradient estimates for Cheeger-harmonic functions imply local isoperimetric inequalities.     

Isoperimetric inequality for the polydisk

Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E ^r (Ω ⁿ ) be the holomorphic Smirnov class on Ω ⁿ with index r. We show that the generalized isoperimetric inequality

$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $

holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.

     

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.     

Isoperimetric inequality for torsional rigidity in multidimensional domains

We consider the Saint-Venant functional P for the torsional rigidity in an arbitrary plane or spatial domain. The main result of this paper is the sharp estimate P ?? (4/n)m, where n is the dimension of the space, and m is the harmonic mean of moments of inertia of the domain about the coordinate planes. The extremal domains are ellipsoids of a special kind. Thus, we obtain a generalization of the isoperimetric inequality proved by E. Nicolay for the torsional rigidity of simply connected plane domains.     

Numerical solutions of the poisson equation

In this article we shall give practical and numerical solutions of the Poisson equation on multidimensional spaces and show their numerical experiments by using computers.     

Isoperimetric inequality and the poincare inequality for distributions of polynomials on convex compact set

An isoperimetric inequality and a Poincare-type inequality are proved for probability measures on the line that are the images of a uniform distribution on a convex compact subset of R ⁿ under polynomial mappings of fixed degree d.     

Isoperimetric inequalities for submanifolds with bounded mean curvature

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.     

Isoperimetric inequalities for submanifolds with bounded mean curvature

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.     

Isoperimetric inequality and equilibrium states of a two-phase medium

A two-sided estimate for the volume of each phase of a two-phase equilibrium state is derived provided that the equilibrium displacement field of the one-phase problem is smooth. The obtained estimate is uniform with respect to the temperature and surface tension coefficient. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 81–88.     

Isoperimetric inequality for radial probability measures on Euclidean spaces

We generalize the Poincaré limit which asserts that the n-dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first n-coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps. Using this generalization, we derive a Gaussian isoperimetric inequality for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.     

Hoeffding's inequality for Markov processes via solution of Poisson's equation

We investigate Hoeffding's inequality for both discrete-time Markov chains and continuous-time Markov processes on a general state space. Our results relax the usual aperiodicity restriction in the literature, and the explicit upper bounds in the inequalities are obtained via the solution of Poisson's equation. The results are further illustrated with applications to queueing theory and reective diffusion processes.     

Estimates of the conformal scalar curvature equation via the method of moving planes

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4). © 1997 John Wiley & Sons, Inc.     

Harnack inequality for the Lagrangian mean curvature flow

If is a Lagrangian manifold immersed into a K?hler-Einstein manifold, nothing is known about its behavior under the mean curvature flow. As a first result we derive a Harnack inequality for the mean curvature potential of compact Lagrangian immersions immersed into . Received March 16, 1997 / Accepted April 24, 1998     

Observability inequality for the Petrovsky equation

The aim of this paper is to study the exact controllability of the Petrovsky equation. Under some checkable geometric assumptions, we establish the observability inequality via the multiplier method for the Dirichlet control problem.