Rearrangements in Carnot Groups

In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a giv     

On a nonlocal elliptic problem arising in the confinement of a plasma in a current carrying stellarator

This paper deals with a nonlocal free boundary problem arising in the study of the dynamics of the confinement of a plasma in a Stellarator device. The free boundary represents the separation between the plasma and vacuum regions, and the nonlocal term involves the notions of relative rearrangement and monotone rearrangement. We establish some new properties on the decreasing rearrangement that can be used to prove the convergence of the approximate problem, and then prove the existence of solutions by Galerkin method. Copyright © 2013 John Wiley & Sons, Ltd.     

An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a Stellarator

We prove the existence and the regularity of weak solutions of a nonlocal elliptic-parabolic free-boundary problem involving the notions of relative rearrangement and monotone rearrangement. The problem arises in the study of the dynamics of a magnetically confined fusion plasma in a Stellarator device when the dimensional analysis on the characteristic times suggests to neglect the inertial acceleration in presence of a time dependent magnetic field.     

Rearrangements in Real Estate Investments

In this note we consider an investment problem in real estate. We show that it can be formulated in terms of a constrained optimization problem, and this leads to a linear rearrangement optimization problem. We address existence, uniqueness, and symmetry of the optimal solution.     

本刊英文版Vol.35(2019),No.7论文摘要（英文）

正Rearrangements in Carnot Groups Juan J.MANFREDI Virginia N.VERA DE SERIO Abstract In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups.We define rearrangement with respect to a given family of     

Absolutely Continuous Embeddings of Rearrangement-Invariant Spaces

Compactness of embeddings between rearrangement invariant spaces is closely related to absolute continuity of these embeddings. We study absolutely continuous embeddings between rearrangement invariant spaces. In particular it is shown that an absolutely continuous embedding is never optimal. We give sufficient (and under additional hypotheses necessary) conditions for absolute continuity of these embeddings. We also provide quantitative estimates of absolutely continuous embeddings.     

Orthogonal Elements in Nonseparable Rearrangement Invariant Spaces

Doklady Mathematics - Let E be a nonseparable rearrangement invariant space, and let E0 denote the closure of the set of all bounded functions in E. We study elements of E orthogonal to the...     

Rearrangements with supporting trees, isomorphisms and shift operators

We study rearrangement operators which admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension to ${L^p_E}$ , where E is a UMD space. We prove the existence of a large subspace ${X_p\subset L^p}$ on which a bounded rearrangement operator acts as an isomorphism. For a class of special shift operators, we construct supporting trees explicitly by combinatorial means.     

Rearrangement transformations on general measure spaces

For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.     

Symmetrization inequalities in the fractional case and Besov embeddings

We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. As a consequence we determine new results for rearrangement invariant hulls of generalized Besov spaces.     

Sequences of Independent Functions in Rearrangement Invariant Spaces

Siberian Mathematical Journal - We obtain some new estimates that show the extremality of the Rademacher system in the set of sequences of independent functions considered in rearrangement...     

Rearrangement formulas for the singular Henkin-Ramirez integral in strictly pseudoconvex domains

We consider the rearrangement and composition formulas for singular Henkin-Ramirez integral in strictly pseudoconvex domains for Cauchy and Kerzman-Stein principal values. These formulas are different.     

Lacunary Walsh series in rearrangement invariant spaces

We prove that the classical results by Rodin and Semenov and by Lindenstrauss and Tzafriri on the subspace generated by the Rademacher system in rearrangement invariant spaces also hold for lacunary Walsh series.     

A Variational Problem for Steady Vortices in a Shear Flow Past an Obstacle

We prove existence of maximizers for a variational problem for a steady vortex anomaly of bounded extent in a shear flow, past an obstacle, in a planar exterior domain. Kinetic energy is maximized subject to the vorticity being a rearrangement of a prescribed function.     

Existence of an extremal ground state energy of a nanostructured quantum dot

This paper is concerned with two rearrangement optimization problems. These problems are motivated by two eigenvalue problems which depend nonlinearly on the eigenvalues. We consider a rational and a quadratic eigenvalue problem with Dirichlet’s boundary condition and investigate two related optimization problems where the goal function is the corresponding first eigenvalue. The first eigenvalue in the rational eigenvalue problem represents the ground state energy of a nanostructured quantum dot. In both the problems, the admissible set is a rearrangement class of a given function.     

A constrained variational problem for steady vortices in a shear flow

We prove existence of maximisers for a variational problem for a steady vortex anomaly of bounded extent in a uniform shear flow in IR². Kinetic energy is maximised subject to the vorticity being a rearrangement of a prescribled function, and subject to a linear pseudo impulse constrint     

Total curvature and rearrangements

We study to what extent rearrangements preserve the integrability properties of higher order derivatives. It is well known that the second order derivatives of the rearrangement of a smooth function are not necessarily inL ¹. We obtain a substitute for this fact. This is done by showing that the total curvature for the graph of the rearrangement of a function is bounded by the total curvature for the graph of the function itself. This posthumous paper was prepared for publication by Vilhelm Adolfsson and Peter Kumlin. The author was supported by a grant from the Swedish Natural Science Research Council.     

On the necessity of the assumptions used to prove Hardy–Littlewood and Riesz rearrangement inequalities

We prove that supermodularity is a necessary condition for the generalized Hardy–Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz-type integral.     

Strictly singular and power-compact operators on Banach lattices

Compactness of the iterates of strictly singular operators on Banach lattices is analyzed. We provide suitable conditions on the behavior of disjoint sequences in a Banach lattice, for strictly singular operators to be Dunford-Pettis, compact or have compact square. Special emphasis is given to the class of rearrangement invariant function spaces (in particular, Orlicz and Lorentz spaces). Moreover, examples of rearrangement invariant function spaces of fixed arbitrary indices with strictly singular non power-compact operators are also presented.     

Rearrangements and radial graphs of constant mean curvature in hyperbolic space

We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques. The second author was partially supported by NSF grant DMS 0603707.