Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

Sobolev and isoperimetric inequalities with monomial weights

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

By using optimal mass transport theory we prove a sharp isoperimetric inequality in $${\textsf {CD}} (0,N)$$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

Affine vs. Euclidean isoperimetric inequalities

Given a positive lower semi-continuous density f on mathbb {R}^2 the weighted volume V_f:=fmathscr {L}^2 is defined on the mathscr {L}^2-measurable sets in mathbb {R}^2. The f-weighted perimeter of a set of finite perimeter E in mathbb {R}^2 is written P_f(E). We study minimisers for the weighted isoperimetric problem If(v):=inf{Pf(E):Eis a set of finite perimeter inR2andVf(E)=v}\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} I_f(v):=\\inf \\Big \\{ P_f(E):E\\text { is a set of finite perimeter in }\\mathbb {R}^2\\text { and }V_f(E)=v\\Big \\} \\end{aligned}$$\\end{document}for v>0. Suppose f takes the form f:mathbb {R}^2rightarrow (0,+infty );xmapsto e^{h(|x|)} where h:[0,+infty )rightarrow mathbb {R} is a non-decreasing convex function. Let v>0 and B a centred ball in mathbb {R}^2 with V_f(B)=v. We show that B is a minimiser for the above variational problem and obtain a uniqueness result.

The theory of sets of finite perimeter and BV functions in Wiener spaces, i.e., Banach spaces endowed with a Gaussian Borel probability measure γ, was initiated by Fukushima and Hino in [9, 10, 11], and has been further investigated in [12, 1, 2, 3]. The basic question one would like to consider is the research of infinite-dimensional analogues of the classical fine properties of BV functions and sets of finite perimeter in finite-dimensional spaces. The class of sets of finite Gaussian perimeter E in a Gaussian Banach space (X, γ) is defined by the integration by parts formula ˆ

The helicity of a smooth vector field defined on a domain in three-space is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. The isoperimetric problem in this setting is to maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the boundary of all domains of given volume in three-space. The Biot–Savart operator starts with a divergence-free vector field defined on and tangent to the boundary of a domain in three-space, regards it as a distribution of electric current, and computes its magnetic field. Restricting the magnetic field to the given domain, we modify it by subtracting a gradient vector field so as to keep it divergence-free while making it tangent to the boundary of the domain. The resulting operator, when extended to the L2 completion of this family of vector fields, is compact and self-adjoint, and thus has a largest eigenvalue, whose corresponding eigenfields are smooth by elliptic regularity. The isoperimetric problem for this modified Biot–Savart operator is to maximize its largest eigenvalue among all domains of given volume in three-space. The curl operator, when restricted to the image of the modified Biot–Savart operator, is its inverse, and the isoperimetric problem for this restriction of the curl is to minimize its smallest positive eigenvalue among all domains of given volume in three-space. These three isoperimetric problems are equivalent to one another. In this paper, we will derive the first variation formulas appropriate to these problems, and use them to constrain the nature of any possible solution. For example, suppose that the vector field V, defined on the compact, smoothly bounded domain Ω, maximizes helicity among all divergence-free vector fields of given nonzero energy, defined on and tangent to the boundary of all such domains of given volume. We will show that (1) |V| is a nonzero constant on the boundary of each component of Ω; (2) all the components of ∂Ω are tori; and (3) the orbits of V are geodesics on ∂Ω. Thus, among smooth simply connected domains, none are optimal in the above sense. In principal, one could have a smooth optimal domain in the shape, say, of a solid torus. However, we believe that there are no smooth optimal domains at all, regardless of topological type, and that the true optimizer looks like the singular domain presented in this paper, which we can think of either as an extreme apple, in which the north and south poles have been pressed together, or as an extreme solid torus, in which the hole has been shrunk to a point. A computational search for this singular optimal domain and the helicity-maximizing vector field on it is at present under way, guided by the first variation formulas in this paper.

In this paper we review some recent results concerning the following non-local isoperimetric problem: Open image in new window where m ∈ (-1,1) is given and prescribes the volume of the two phases {u = 1} and {u = -1}. Here PΩ stands for the perimeter relative to Ω (in the sense of Caccioppoli-De Giorgi) and BV(Ω; {-1, 1}) denotes the space of functions of bounded variation taking values in {-1, 1}. We refer to [4] and [33] for a rather complete account on the properties of functions of bounded variations and sets of finite perimeter.

Steiner symmetrization in n linearly independent directions transforms every compact subset of $\mathbb {R}^{n}$ into a set of finite perimeter.

Symmetrization is one of the most powerful mathematical tools with several applications both in Analysis and Geometry. Probably the most remarkable application of Steiner symmetrization of sets is the De Giorgi proof (see [14], [25]) of the isoperimetric property of the sphere, while the spherical symmetrization of functions has several applications to PDEs and Calculus of Variations and to integral inequalities of Poincare and Sobolev type (see for instance [23], [24], [19], [20]). The two model functionals that we shall consider in the sequel are: the perimeter of a set E in IR and the Dirichlet integral of a scalar function u. It is well known that on replacing E or u by its Steiner symmetral or its spherical symmetrization, respectively, both these quantities decrease. This fact is classical when E is a smooth open set and u is a C function ([22], [21]). Moreover, on approximating a set of finite perimeter with smooth open sets or a Sobolev function by C functions, these inequalities can be easily extended by lower semicontinuity to the general setting ([19], [25], [2], [4]). However, an approximation argument gives no information about the equality case. Thus, if one is interested in understanding when equality occurs, one has to carry on a deeper analysis, based on fine properties of sets of finite perimeter and Sobolev functions. Let us start by recalling what the Steiner symmetrization of a measurable set E is. For simplicity, and without loss of generality, in the sequel we shall always consider the symmetrization of E in the vertical direction. To this aim, it is convenient to denote the points x in IR also by (x′, y), where x′ ∈ IRn−1 and y ∈ IR. Thus, given x′ ∈ IRn−1, we shall denote by Ex′ the corresponding one-dimensional section of E

On an isoperimetric problem with power-law potentials and external attraction

Higher-order Sobolev embeddings and isoperimetric inequalities

In this paper we expand the concept of a really significant probabilistic measure in the case when the measure takes values in the algebra of bihyperbolic numbers. The basic properties of bihyperbolic numbers are given, in particular idempotents, main ideals generated by idempotents, Pierce's decompo\-sition and the set of zero divisors of the algebra of bihyperbolic numbers are determined. We entered the relation of partial order on the set of bihyperbolic numbers, by means of which the bihyperbolic significant modulus is defined and its basic properties are proved. In addition, some bihyperbolic modules can be endowed with a bihyperbolic significant norms that take values in a set of non-negative bihyperbolic numbers. We define $\sigma$-additive functions of sets in a measurable space that take appropriately normalized bihyperbolic values, which we call a bihyperbolic significant probability. It is proved that such a bihyperbolic probability satisfies the basic properties of the classical probability. A representation of the bihyperbolic probability measure is given and its main properties are proved. A bihyperbolically significant random variable is defined on a bihyperbolic probability space, and this variable is a bihyperbolic measurable function in the same space. We proved the criterion of measurability of a function with values in the algebra of bihyperbolic numbers, and the basic properties of bihyperbolic random variables are formulated and proved. Special cases have been studied in which the bihyperbolic probability and the bihyperbolic random variable take values that are zero divisors of bihyperbolic algebra. Although bihyperbolic numbers are less popular than hyperbolic numbers, bicomplex numbers, or quaternions, they have a number of important properties that can be useful, particularly in the study of partial differential equations also in mathematical statistics for testing complex hypotheses, in thermodynamics and statistical physics.

Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities