On the isoperimetric problem in Euclidean space with density (original) (raw)

Some Isoperimetric Problems in Planes with Density

Journal of Geometric Analysis, 2010

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in R 2 . Finally, we give a list of related open questions.

Isoperimetric comparison theorems for manifolds with density

Calculus of Variations and Partial Differential Equations, 2009

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.

Symmetry of minimizers of a Gaussian isoperimetric problem

Probability Theory and Related Fields

We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. The second term is in competition with the perimeter, balancing the mass with respect to the origin, and because of that the solution is not always the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the barycenter term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one. As another corollary we obtain the optimal constant in the quantitative Gaussian isoperimetric inequality.

On isoperimetric sets of radially symmetric measures

Concentration, Functional Inequalities and Isoperimetry, 2011

We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. It is shown, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. In addition, we establish some comparison results for general log-convex measures.

Balls Isoperimetric in R^n with Volume and Perimeter Densities r^m and r^k

2016

We have discovered a "little" gap in our proof of the sharp conjecture that in R^n with volume and perimeter densities r^m and r^k, balls about the origin are uniquely isoperimetric if 0 < m ≤ k - k/(n+k-1), that is, if they are stable (and m > 0). The implicit unjustified assumption is that the generating curve is convex.

Some sharp isoperimetric theorems for Riemannian manifolds

Indiana University Mathematics Journal, 2000

We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using differential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.

On the isoperimetric deficit in Gauss space

American Journal of Mathematics, 2011

We find a sharp quantitative estimate for the isoperimetric inequality in R n with the Gaussian measure. Contents 1. Introduction 1 2. Background and notation 4 3. The one-dimensional case 6 4. A geometric proof of the Gaussian isoperimetric theorem 10 5. Reduction to (n − 1)-symmetric sets 18 6. The two-dimensional case 24 7. The n-dimensional case 37 References 41