Quantitative Isoperimetric Inequality Research Articles

On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry

On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry

Uniform bound on the number of partitions for optimal configurations of the Ohta–Kawasaki energy in 3D

AbstractWe study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $\Omega\subset\mathbb{R}^n$ open, bounded, and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds that $\lambda\_1(\Omega)-\lambda\_1(B)\geq c(P(\Omega)-P(B))^2$, where $\lambda\_1(\cdot)$ denotes the first Dirichlet eigenvalue of a set and $P(\cdot)$ its perimeter. The heart of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter.

A Poincaré type inequality with three constraints

In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$\begin{aligned}\inf _{u\in {\mathcal {W}}} \frac{\displaystyle \int _{-\pi }^{\pi }[(u')^2-u^2]d\theta }{\displaystyle \left[ \int _{-\pi }^{\pi } |u| d\theta \right] ^2} \end{aligned}$$ where a function $$u\in {\mathcal {W}}$$ is a $$H^1(-\pi ,\pi )$$ periodic function, with zero average on $$(-\pi ,\pi )$$ and orthogonal to sine and cosine.

The Riemannian quantitative isoperimetric inequality

We study the Riemannian quantitive isoperimetric inequality. We show that a direct analogue of the Euclidean quantitative isoperimetric inequality is—in general—false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Łojasiewicz–Simon inequality. The main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.

Quantitative estimates for the Bakry–Ledoux isoperimetric inequality

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\operatorname{Ric}\_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag’s needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincaré inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.

A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is 1∕2 and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

A weighted quantitative isoperimetric inequality for Korányi sphere in Heisenberg group ℍ^n

A weighted quantitative isoperimetric inequality for Korányi sphere in Heisenberg group ℍ^n

Symmetry and isoperimetry for Riemannian surfaces

For a domain \(\Omega \) in a geodesically convex surface, we introduce a scattering energy \(\mathcal {E}(\Omega )\), which measures the asymmetry of \(\Omega \) by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving \(\mathcal {E}(\Omega )\) and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new of the sharp Sobolev inequality for Riemannian surfaces.

A quantitative Weinstock inequality for convex sets

This paper is devoted to the study of a quantitative Weinstock inequality in higher dimension for the first non trivial Steklov eigenvalue of the Laplace operator for convex sets. The key role is played by a quantitative isoperimetric inequality which involves the boundary momentum, the volume and the perimeter of a convex open set of $${\mathbb {R}}^n$$, $$n \ge 2$$.

Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood

In this paper we extend the results of A strong minimax property of nondegenerate minimal submanifolds, by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of A selection principle for the sharp quantitative isoperimetric inequality, by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands.

Weighted quantitative isoperimetric inequalities in the Grushin space {R}^{h+1} with density |x|^{p}

In this paper, we prove weighted quantitative isoperimetric inequalities for the set E_{alpha}= {(x,y)in {R}^{h+1}: vert y vert <int_{arcsin vert x vert }^{frac{pi}{2}}sin^{alpha +1}(t),dt, vert x vert <1 } in half-cylinders in the Grushin space {R}^{h+1} with density vert x vert ^{p}, pgeq0.

A minimal partition problem with trace constraint in the Grushin plane

We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the least amount of perimeter, under an additional "one-dimensional" constraint on the intersections of their boundaries. We prove existence of regular solutions for this problem, and we characterize them in terms of isoperimetric sets, showing differences with the Euclidean case. The problem arises from the study of quantitative isoperimetric inequalities and has connections with the theory of minimal clusters.

On the quantitative isoperimetric inequality in the plane

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω , different from a ball, which minimizes the ratio δ ( Ω ) / λ 2 ( Ω ), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

Existence and properties of certain critical points of the Cahn-Hilliard energy

The Cahn-Hilliard energy landscape on the torus is explored in the critical regime of large system size and mean value close to $-1$. Existence and properties of a droplet-shaped local energy minimizer are established. A standard mountain pass argument leads to the existence of a saddle point whose energy is equal to the energy barrier, for which a quantitative bound is deduced. In addition, finer properties of the local minimizer and appropriately defined constrained minimizers are deduced. The proofs employ the $\Gamma$-limit (identified in a previous work), quantitative isoperimetric inequalities, variational arguments, and Steiner symmetrization.

A sharp quantitative version of Hales' isoperimetric honeycomb theorem

We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.

A quantitative isoperimetric inequality on the sphere

Abstract In this paper we prove a quantitative version of the isoperimetric inequality on the sphere with a constant independent of the volume of the set E.

A Strong Form of the Quantitative Wulff Inequality

Quantitative isoperimetric inequalities are shown for anisotropic surface energies where the isoperimetric deficit controls both the Fraenkel asymmetry and a measure of the oscillation of the boundary with respect to the boundary of the corresponding Wulff shape.

The quantitative isoperimetric inequality and related topics

We present some recent stability results concerning the isoperimetric inequality and other related geometric and functional inequalities. The main techniques and approaches to this field are discussed.

Quantitative isoperimetric inequalities in $$\mathbb {H}^n$$ H n

In the Heisenberg group H^n, we prove quantitative isoperimetric inequalities for Pansu's spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.