Sobolev type inequalities for rearrangement invariant spaces

In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient) are well understood. By means of an alternative functional, we obtain new Sobolev inequalities which are finer than (and not necessarily equivalent to) the ones mentioned above.     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC     

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.     

Isoperimetric inequalities of Euclidean type in metric spaces

No Abstract. .Received: August 2003 Revision: June 2004 Accepted: August 2004     

Ends of metric measure spaces and Sobolev inequalities

Generalizing work of Li and Wang, we prove sharp volume growth/decay rates for ends of metric measure spaces supporting a (p,p)-Sobolev inequality. A sharp result for (q,p)-Sobolev inequalities is also proved. The first author was partially supported by Enterprise Ireland.     

Characterizations of Sobolev inequalities on metric spaces

We discuss Maz'ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces.     

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardy?s inequality in a limiting case are also considered.     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

A mass transportation approach for Sobolev inequalities in variable exponent spaces

In this paper we provide a proof of the Sobolev–Poincaré inequality for variable exponent spaces by means of mass transportation methods, in the spirit of Cordero-Erausquin et al. (Adv Math 182(2):307–332, 2004). The importance of this approach is that the method is flexible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result of Fan (J Math Anal Appl 339(2):1395–1412, 2008) by obtaining an explicit dependence of the exponent in the constant.     

Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ball B ⁴ with the following properties: (i) B ⁴ has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B ⁴.     

Some existence and uniqueness theorems for doubly periodic minimal surfaces

Oblatum 5-XI-1991 & 13-II-1992     

Isoperimetric inequalities for polarization and virtual mass

This paper is the continuation of a previous investigation. Here representations for the components of the virtual mass and polarization tensors in terms of boundary integrals are given and new isoperimetric inequalities for star-shaped bodies are derived. Research supported by NSF Grant, MCS 8300842. Research supported by Natural Sciences and Engineering Research Council Canada.     

On the existence of minimal surfaces with singular boundaries

In 1931, Jesse Douglas showed that in , every set of rectifiable Jordan curves, with , bounds an area-minimizing minimal surface with prescribed topological type if a ``condition of cohesion' is satisfied. In this paper, it is established that under similar conditions, this result can be extended to non-Jordan curves.

     

On the existence of solutions in weighted Sobolev spaces for the exterior Oseen problem

In this Note, we present existence and uniqueness results for the exterior Oseen problem. In order to control the behavior at infinity of functions, we use as functional framework weighted Sobolev spaces. The results rely on a $L^p$-theory for $1<p<\infty$. To cite this article: C. Amrouche, U. Razafison, C. R. Acad. Sci. Paris, Ser. I 341 (2005).