An evolution variational inequality related to a diffusion-absorption problem

An implicit non-steady free boundary problem is transformed into a variational inequality, which is solved by means of a semi-discretization technique.     

An isoperimetric inequality for the first eigenfunction in the fixed membrane problem

Summary An isoperimetric inequality is obtained which relates theL ₁-andL ₂-integrals of the first eigenfunction in the problem of the vibrating clamped membrane.

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Zusammenfassung Für das Problem der eingespannten schwingenden Membran wird zwischen demL ₁-und demL ₂-Integral der ersten Eigenfunktion eine isoperimetrische Ungleichung hergeleitet.

     

An isoperimetric inequality for the Wiener sausage

Let (ξ(s)) _{s?≥ 0} be a standard Brownian motion in d?≥ 1 dimensions and let (D _s ) _{s ≥?0} be a collection of open sets in ${\mathbb{R}^d}$ . For each s, let B _s be a ball centered at 0 with vol(B _s ) =?vol(D _s ). We show that ${\mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + D_s))] \geq \mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + B_s))]}$ , for all t. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.     

An isoperimetric inequality for the Heisenberg groups

We show that the Heisenberg groups of dimension five and higher, considered as Riemannian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area ~ L ².) This implies several important results about isoperimetric inequalities for discrete groups that act either on or on complex hyperbolic space, and provides interesting examples in geometric group theory. The proof consists of explicit construction of a disk spanning each loop in . Submitted: April 1997, Final version: November 1997     

An isoperimetric problem for tetrahedra

It is proved that a regular tetrahedron has the maximal possible surface area among all tetrahedra having surface with unit geodesic diameter. An independent proof of O’Rourke-Schevon’s theorem about polar points on a convex polyhedron is given. A. D. Aleksandrov’s general problem on the area of a convex surface with fixed geodesic diameter is discussed. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 28–55.     

Stability of a reverse isoperimetric inequality

In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303-308], which states that if K is a convex domain in the plane with perimeter p(K) and area a(K), then one gets , where denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc.     

The isoperimetric inequality on a surface

We prove a new isoperimetric inequality which relates the area of a multiply connected curved surface, its Euler characteristic, the length of its boundary, and its Gaussian curvature. Received: 31 July 1998     

An extremal problem related to Kolmogoroff’s inequality for bounded functions

LetA andB be positive numbers andm andn positive integers,m. Then there is for complex valued functions φ onR with sufficient differentiability and boundedness properties a representation wherev ₁ andv ₂ are bounded Borel measures withv ₁ absolutely continuous, such that there exists a function φ with ∣φ⁽ⁿ⁾∣ ?A and ∣φ∣ ?A onR and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.     

On the linear isoperimetric inequality

This paper contains a sharp version of the well-known linear isoperimetric inequality for minimal surfacesX area(X)1/2oscillation(X)length(X).Supported by Sonderforschungsbereich 72 der Deutschen Forschungsgemeinschaft at Bonn University.     

An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in {\mathbb C}^2 related to Oh's conjecture

We compute loops integrals on Hamiltonian stationary Lagrangian tori in which are symplectic invariants, then we show an isoperimetric inequality involving these invariants and the area. Finally, we show that the flat torus has least area among Hamiltonian stationary Lagrangian tori of its isotopy class. Received: 4 December 2000; in final form: 18 January 2002 / Published online: 5 September 2002     

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

Petty's projection inequality and Santalo's affine isoperimetric inequality

Using results of K. Kiener and the Riesz-Sobolev convolution inequality we give a new proof of Petty's projection inequality. By the same method we also obtain a proof of Santalo's affine isoperimetric inequality.Supported in part by BSF and Erwin Schrödinger Auslandsstipendium J0630, J0804.     

On a covering problem related to the centered Hardy-Littlewood maximal inequality

We find the exact value of the best possible constant associated with a covering problem on the real line.     

On the isoperimetric inequality on a minimal surface

We derive an explicit formula for the isoperimetric defect of an arbitrary minimal surface ,in terms of a double integral over the surface of certain geometric quantities, together with a double boundary integral which always has the correct sign. As a by-product of these computations we show that the best known universal isoperimetric estimate, that for any minimal surface (due to L. Simon), may be improved to the universal estimate .Received: 21 June 2001, Accepted: 16 June 2002, Published online: 5 September 2002