Relative Isoperimetric Inequality Research Articles

The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in [10] and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in [5], provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in [11]. Furthermore, we use this estimate and the idea of Brendle in his recent work [3] to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in Rn+m, which is optimal and gives an affirmative answer to an open problem proposed by Choe in [9], Open Problem 12.6, when the codimension m≤2.

Total positive curvature and the equality case in the relative isoperimetric inequality outside convex domains

We settle the case of equality for the relative isoperimetric inequality outside any arbitrary convex set with not empty interior.

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

AbstractLet $$0<\alpha <d$$ 0 < α < d and $$1\le p<d/\alpha $$ 1 ≤ p < d / α . We present a proof that for all $$f\in W^{1,p}({\mathbb {R}}^d)$$ f ∈ W 1 , p ( R d ) both the centered and the uncentered Hardy–Littlewood fractional maximal operator $${\mathrm {M}}_\alpha f$$ M α f are weakly differentiable and $$ \Vert \nabla {\mathrm {M}}_\alpha f\Vert _{p^*} \le C_{d,\alpha ,p} \Vert \nabla f\Vert _p , $$ ‖ ∇ M α f ‖ p ∗ ≤ C d , α , p ‖ ∇ f ‖ p , where $$ p^* = (p^{-1}-\alpha /d)^{-1} . $$ p ∗ = ( p - 1 - α / d ) - 1 . In particular it covers the endpoint case $$p=1$$ p = 1 for $$0<\alpha <1$$ 0 < α < 1 where the bound was previously unknown. For $$p=1$$ p = 1 we can replace $$W^{1,1}({\mathbb {R}}^d)$$ W 1 , 1 ( R d ) by $$\mathrm {BV}({\mathbb {R}}^d)$$ BV ( R d ) . The ingredients used are a pointwise estimate for the gradient of the fractional maximal function, the layer cake formula, a Vitali type argument, a reduction from balls to dyadic cubes, the coarea formula, a relative isoperimetric inequality and an earlier established result for $$\alpha =0$$ α = 0 in the dyadic setting. We use that for $$\alpha >0$$ α > 0 the fractional maximal function does not use certain small balls. For $$\alpha =0$$ α = 0 the proof collapses.

Isoperimetric and Poincaré Inequalities on Non-Self-Similar Sierpiński Sponges: the Borderline Case

Abstract In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.

A sharp relative isoperimetric inequality for the square

We compute the exact value of the least “relative perimeter” of a shape S, with a given area, contained in a unit square; the relative perimeter of S being the length of the boundary of S that does not touch the border of the square.

A weighted relative isoperimetric inequality in convex cones

A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation. The method improves several inequalities in the literature, e.g. constants in a theorem of Cabre--Ros--Oton--Serra. Applications are given in the context of a generalization of the log-convex density conjecture due to Brakke and resolved by Chambers: in the case of $\alpha-$homogeneous ($\alpha>0$), concave densities, (mod translations) balls centered at the origin and intersected with the cone are proved to uniquely minimize the weighted perimeter with a weighted mass constraint. In particular, if the cone is taken to be $\{x_n>0\}$, reflecting the density, balls intersected with $\{x_n>0\}$ remain (mod translations) unique minimizers in the $\mathbb{R}^n$ analog in the case when the density vanishes on $\{x_n=0\}$.

Weighted Cheeger sets are domains of isoperimetry

We consider a generalization of the Cheeger problem in a bounded, open set $\Omega$ by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer $A$ of this weighted Cheeger problem such that $H^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric inequality. If $\Omega$ itself is a connected minimizer such that $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$, then it allows the classical Sobolev and $BV$ embeddings and the classical $BV$ trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and $\Omega$ is such that $|\partial \Omega|=0$ and $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$.

Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights

We consider a stationary and ergodic random field $\{\omega(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $\omega(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(\omega)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality.

A new multicomponent Poincaré–Beckner inequality

We prove a new vectorial functional inequality of Poincaré–Beckner type. The inequality may be interpreted as an entropy–entropy production one for a gradient flow in the metric space of Radon measures. The proof uses subtle analysis of combinations of related super- and sub-level sets employing the coarea formula and the relative isoperimetric inequality.

Weakly differentiable functions on varifolds

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincar\'e type embeddings, embeddings into spaces of continuous and sometimes H\"older continuous functions, pointwise differentiability results both of approximate and integral type as well as coarea formulae. As prerequisite for this study decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose. As applications the finiteness of the geodesic distance associated to varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.

Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces

We study equivalence between the Poincaré inequality and several different relative isoperimetric inequalities on metric measure spaces. We then use these inequalities to establish sufficient conditions for the finite perimeter of sets.

Relative isoperimetric inequalities for minimal submanifolds outside a convex set

AbstractLet C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ∩∂C and ∂Σ ∼ ∂C is radially connected from a point p ∈ ∂Σ∩∂C. We introduce a modified volume Mp(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.

A relative isoperimetric inequality for certain warped product spaces

Given a warped product space ℝ ×f N with logarithmically convex warping function f, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a k-dimensional region F ⊂ {b} × N , and the horizontal graph C ⊂ ℝ ×f N of an almost everywhere differentiable map over F, we prove that the k-volume of C is always at least the k-volume of the smooth constant height graph over F that traps the same (1 + k)-volume above F as C. We use this to solve a Dido problem for graphs over vertical fibers, and show that, if the warping function is unbounded on the set of horizontal values above a vertical fiber, the volume trapped above that fiber by a graph C is no greater than the k-volume of C times a constant that depends only on the warping function.

A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

The relative isoperimetric inequality inside an open, convex cone $\mathcal{C}$ states that, at fixed volume, $B_{r} \cap\mathcal{C}$ minimizes the perimeter inside $\mathcal{C}$ . Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov’s proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal{C}$ . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167–211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.

The relative isoperimetric inequality on a conformally parabolic manifold with boundary

For an arbitrary noncompact n-dimensional Riemannian manifold with a boundary of conformally parabolic type it is proved that there exists a conformal change of metric such that a relative isoperimetric inequality of the same form as in the closed n-dimensional Euclidean half-space holds on the manifold with the new metric. This isoperimetric inequality is asymptotically sharp.Bibliography: 6 titles.

Acharacterization of Newtonian functions with zero boundary values

We give an intrinsic characterization of the property that the zero extension of a Newtonian function, defined on an open set in a doubling metric measure space supporting a strong relative isoperimetric inequality, belongs to the Newtonian space on the entire metric space. The theory of functions of bounded variation is used extensively in the argument and we also provide a structure theorem for sets of finite perimeter under the assumption of a strong relative isoperimetric inequality. The characterization is used to prove a strong version of quasicontinuity of Newtonian functions.

Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold

Let be a domain on an -dimensional minimal submanifold in the outside of a convex set in or . The modified volume is introduced by Choe and Gulliver (1992) and we prove a sharp modified relative isoperimetric inequality for the domain , , where is the volume of the unit ball of . For any domain on a minimal surface in the outside convex set in an -dimensional Riemannian manifold, we prove a weak relative isoperimetric inequality , where is an upper bound of sectional curvature of the Riemannian manifold.

Area-minimizing regions with small volume in Riemannian manifolds with boundary

Given a domain Ω of a Riemannian manifold, we prove that regions minimizing the area (relative to Ω) are nearly the maxima of the mean curvature of ∂Ω when their volume tends to zero. We deduce some sharp local relative isoperimetric inequalities involving mean curvature comparisons.

A class of degenerate elliptic equations and a Dido's problem with respect to a measure

In this paper we consider the following class of linear elliptic problems { − div ( A ( x ) ∇ u ) = x N k exp ( − | x | 2 2 ) f ( x ) in Ω , u = 0 on ∂ Ω ∖ { x N = 0 } , where k ⩾ 0 , Ω is a domain (possibly unbounded) of R + N = { x = ( x 1 , … , x N ) ∈ R N : x N > 0 } , f belongs to a suitable weighted Lebesgue space and A ( x ) = ( a i j ( x ) ) i j is a symmetric matrix with measurable coefficients satisfying x N k exp ( − | x | 2 2 ) | ζ | 2 ⩽ a i j ( x ) ζ i ζ j ⩽ C x N k exp ( − | x | 2 2 ) | ζ | 2 . We compare the solution to such a problem with the solution to a symmetric one-dimensional problem belonging to the same class. Our approach use classical symmetrization methods adapted to a relative isoperimetric inequality with respect to a measure related to the structure of the equation.

Hyperplane sections of convex bodies

We prove sharp inequalities for the volumes of hyperplane sections bisecting a convex body in R n . This leads to a relative isoperimetric inequality for arbitrary hyperplane sections of a convex body.