Sharp convex Lorentz–Sobolev inequalities

New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L ^p Minkowski problem. New L ^p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.     

Regularity for minimal boundaries in R n ¶with mean curvature in L n

Let be a minimal set with mean curvature in L ⁿ that is a minimum of the functional , where is open and . We prove that if then can be parametrized over the (n−1)-dimensional disk with a C ^α mapping with C ^α inverse. Received: 11 July 1997 / Revised version: 24 February 1998     

Isoperimetric properties of Euclidean boundary moments of a simply connected domain

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L ^p -norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.     

Isoperimetric inequality from the poisson equation via curvature

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q‐regular measure, where Q > 1, that supports a local L²‐Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L^∞‐norm of the gradient Du can be bounded by the Lorentz norm L^Q,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on $\||Du|\|_{L^\infty_{\rm loc}}$ . © 2011 Wiley Periodicals, Inc.     

Galerkin method for a Stefan-type problem in one space dimension

Based on a Landau-type transformation, both continuous and discrete in time L²-Galerkin methods are applied to a single-phase Stefan-type problem in one space dimension. Optimal rates of convergence in L^ℵ, L^∞, and H¹-norms are derived and computational results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 393–416, 1997     

L ∞ Variational Problems with Running Costs and Constraints

Various approaches are used to derive the Aronsson–Euler equations for L ^∞ calculus of variations problems with constraints. The problems considered involve holonomic, nonholonomic, isoperimetric, and isosupremic constraints on the minimizer. In addition, we derive the Aronsson–Euler equation for the basic L ^∞ problem with a running cost and then consider properties of an absolute minimizer. Many open problems are introduced for further study.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poincaré inequality for general symmetric forms

Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.     

Anchored expansion and random walk

This paper studies anchored expansion, a non-uniform version of the strong isoperimetric inequality. We show that every graph with i-anchored expansion contains a subgraph with isoperimetric (Cheeger) constant at least i. We prove a conjecture by Benjamini, Lyons and Schramm (1999) that in such graphs the random walk escapes with a positive lim inf speed. We also show that anchored expansion implies a heat-kernel decay bound of order exp(—cn ^1/3). Submitted: September 1999, Revision: January 2000.     

Vertex algebras and the formal loop space

We construct a certain algebro-geometric version L(X)\mathcal{L}(X) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme L⁰(X)\mathcal{L}^{0}(X) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on L(X)\mathcal{L}(X) supported in L⁰(X)\mathcal{L}^{0}(X) . We also show that L(X)\mathcal{L}(X) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that all linear constructions applied to the free loop space produce vertex algebras.     

Large deviations and continuum limit in the 2D Ising model

Summary. We study the 2D Ising model in a rectangular box Λ _L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ _t∈ΛL σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m ^* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {|∑ _t∈ΛL σ(t) −m|Λ _L ||≤|Λ _L |L ^{− c} }, with 0<c<1/4 and −m ^*<m<m ^*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type. Received: 17 October 1996 / In revised form: 7 March 1997     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.     

A Strong Converse Inequality for Gamma-Type Operators

We give a strong converse inequality of type A in the usual sup-norm for a noncentered gamma operator L _t ^* , providing at the same time upper and lower constants. This operator, which does not preserve smooth functions, is connected with real Laplace transforms and Poisson mixtures. We use a probabilistic approach based on the representation of L _t ^* in terms of gamma processes. October 15, 1997. Date revised: September 14, 1998. Date accepted: October 7, 1998.     

Isoperimetric quotients for a decomposed convex body

In the Euclidean plane, decompose a convex body T into n\geq 2 convex bodies T ₁ ,\ldots ,T _n with areas also denoted by T ₁ ,\ldots ,T _n , and with perimeters L ₁ ,\ldots ,L _n . For T a polygon with at most six sides, G. Fejes Tóth and also L. Fejes Tóth showed that the isoperimetric quotient (L ₁ + ⋅s + L _n )/(\sqrt T ₁ + ⋅s + \sqrt T _n ) is greater than the corresponding isoperimetric quotient of a regular hexagon if T _i /T _j for any i, j is bounded from below by some appropriate constant. We generalize this result to any convex body T , and we show the analogous result for the isoperimetric quotient (L ² ₁ + ⋅s + L ² _n )/(T ₁ + ⋅s + T _n ) . Received April 21, 1999, and in revised form June 21, 2000. Online publication January 17, 2001.     

Affine symplectic geometry I: Applications to geometric inequalities

We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ³. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic oscillator operator inL ²(M), whereM is a Hadamard manifold. This article is dedicated to my dear friend Julia Rashba     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Isoperimetric monotony of the L p -norm of the warping function of a plane simply connected domain

Let G be a simply connected domain and let u(x,G) be its warping function. We prove that L ^p -norms of functions u and u ^?1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St.Venant-Pólya and the Payne inequalities.     

Best one-sided L 1-approximation by harmonic functions

Let , where B is the open unit ball in (), and let denote the collection of functions h in which are harmonic on B and satisfy on . A function h ^* in is called a best harmonic one-sided L ¹-approximant to f if for all h in . This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L ¹-approximation without a one-sidedness condition. Received: 25 September 1997     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Geometry of kinematicK-loops

AK-loop is called kinematic, if a further condition (K7) is valid. Such a loop (L, ⊕) can be provided in a natural way with a left and right structureL andG such that (L,L) and (L,G) become incidence (linear) spaces. For (L,L) andt∈L, each left translationt ⁺:L→L;x→b⊕x is a collineation and (L,G) can be turned in an incidence space with parallelism (L,G, ‖). Examples of kinematicK-loops are given for which the corresponding automorphisms δ_a,b are either the identity or fixed point free.