- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 17:02:49
13
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=26&iss=3&update_since=2024-03-28
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
26
2010
3
Topological and analytical properties of Sobolev bundles. II. Higher dimensional cases
Takeshi
Isobe
Tokyo Institute of Technology, TOKYO, JAPAN
Sobolev bundle, topology of bundle, Yang-Mills functional, variational problem
We define various classes of Sobolev bundles and connections and study their topological and analytical properties. We show that certain kinds of topologies (which depend on the classes) are well-defined for such bundles and they are stable with respect to the natural Sobolev topologies. We also extend the classical Chern-Weil theory for such classes of bundles and connections. Applications related to variational problems for the Yang-Mills functional are also given.
Functional analysis
Manifolds and cell complexes
Global analysis, analysis on manifolds
General
729
798
10.4171/RMI/616
http://www.ems-ph.org/doi/10.4171/RMI/616
Contact properties of codimension 2 submanifolds with flat normal bundle
Juan José
Nuño-Ballesteros
Universitat de València, BURJASSOT (VALENCIA), SPAIN
M.C.
Romero-Fuster
Universitat de València, BURJASSOT (VALENCIA), SPAIN
Asymptotic directions, ν-principal curvature foliation, umbilicity, sphericity, normal curvature
Given an immersed submanifold $M^n\subset\mathbb{R}^{n+2}$, we characterize the vanishing of the normal curvature $R_D$ at a point $p \in M$ in terms of the behaviour of the asymptotic directions and the curvature locus at $p$. We relate the affine properties of codimension 2 submanifolds with flat normal bundle with the conformal properties of hypersurfaces in Euclidean space. We also characterize the semiumbilical, hypespherical and conformally flat submanifolds of codimension 2 in terms of their curvature loci.
Differential geometry
Global analysis, analysis on manifolds
General
799
824
10.4171/RMI/617
http://www.ems-ph.org/doi/10.4171/RMI/617
Le Théorème du symbole total d'un opérateur différentiel $p$-adique
Zoghman
Mebkhout
Université Paris 7 Denis Diderot, PARIS CEDEX 05, FRANCE
Luis
Narváez Macarro
Universidad de Sevilla, SEVILLA, SPAIN
Affinoid algebra, Dwork-Monsky-Washnitzer algebra, †-scheme, †-adic differential operator
Let ${\mathcal X}^\dagger$ be a smooth $\dagger$-scheme (in the sense of Meredith) over a complete discrete valuation ring $(V, {\mathfrak m})$ of unequal characteristics $(0,p)$ and let ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ be the sheaf of $V$-linear endomorphisms of ${\mathcal O}_{{\mathcal X}^\dagger}$ whose reduction modulo ${\mathfrak m}^s$ is a linear differential operator of order bounded by an affine function in $s$. In this paper we prove that locally there is an ${\mathcal O}_{{\mathcal X}^\dagger}$-isomorphism between the sections of ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ and the overconvergent total symbols, and we deduce a cohomological triviality property.
Algebraic geometry
General
825
859
10.4171/RMI/618
http://www.ems-ph.org/doi/10.4171/RMI/618
The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages
Idris
Assani
University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES
Zoltán
Buczolich
Eötvös Loránd University, BUDAPEST, HUNGARY
Furstenberg averages, bilinear Hardy–Littlewood maximal function
Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $$ R^* : (f, g) \in L^1 \times L^1 \rightarrow R^*(f, g)(x) = \sup_{n} \frac{f(T^n x) g(T^{2n} x)}{n}. $$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy-Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1$. The Furstenberg averages do not converge for all pairs of $(L^1,L^1)$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^p,L^q)$ functions with $\frac{1}{p}+\frac{1}{q} \leq 1$.
Dynamical systems and ergodic theory
Measure and integration
General
861
890
10.4171/RMI/619
http://www.ems-ph.org/doi/10.4171/RMI/619
Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems
Yoshie
Sugiyama
Tsuda University, TOKYO, JAPAN
Parabolic system of degenerate type, Keller-Segel, porous medium, Aronson- Bénilan estimate, interface, optimal Hölder continuity
We consider the Keller-Segel system of degenerate type (KS)$_m$ with $m > 1$ below. We establish a uniform estimate of $\partial_x^2 u^{m-1}$ from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)$_m$. In addition, we find that the set $D(t):=\{ x \in \mathbb{R}; u(x,t) > 0\}$ of positive region to the solution $u$ is monotonically non-decreasing with respect to $t$.
Partial differential equations
General
891
913
10.4171/RMI/620
http://www.ems-ph.org/doi/10.4171/RMI/620
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities
Qionglei
Chen
Institute of Applied Physics and Computational Mathematics, BEIJING, HAIDIAN DISTRICT, CHINA
Changxing
Miao
Institute of Applied Physics and Computational Mathematics, BEIJING, CHINA
Zhifei
Zhang
Peking University, BEIJING, CHINA
Compressible Navier-Stokes equations, Besov spaces, Bony’s paraproduct, Fourier localization
In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.
Partial differential equations
General
915
946
10.4171/RMI/621
http://www.ems-ph.org/doi/10.4171/RMI/621
On some maximal multipliers in $L^p$
Ciprian
Demeter
Indiana University, BLOOMINGTON, UNITED STATES
Maximal multipliers, phase space projections
We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1 < p < \infty$.
Fourier analysis
General
947
964
10.4171/RMI/622
http://www.ems-ph.org/doi/10.4171/RMI/622
Overdetermined problems in unbounded domains with Lipschitz singularities
Alberto
Farina
Université Picardie Jules Verne, AMIENS, FRANCE
Enrico
Valdinoci
Università di Roma Tor Vergata, ROMA, ITALY
Elliptic partial differential equations, rigidity results, nonexistence of solutions
We study the overdetermined problem $$ \left\{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\Omega$,} \\ u = 0 & \mbox{ on $\partial\Omega$,} \\ \partial_\nu u = c & \mbox{ on $\Gamma$,} \end{array} \right. $$ where $\Omega$ is a locally Lipschitz epigraph, that is $C^3$ on $\Gamma\subseteq\partial\Omega$, with $\partial\Omega\setminus\Gamma$ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets $\Omega$ for which monotone solutions exist. In particular, if $\mathcal{C} \in \mathbb{R}^n$ is a cone and either $n=2$ or $n=3$ and $f \ge 0$, then there exists no solution of $$ \left\{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\mathcal{C}$,} \\ u > 0 & \mbox{ in $\mathcal{C}$,} \\ u = 0 & \mbox{ on $\partial\mathcal{C}$,} \\ \partial_\nu u = c & \mbox{ on $\partial\mathcal{C} \setminus \{0\}$.} \end{array} \right. $$ This answers a question raised by Juan Luis Vázquez.
Partial differential equations
General
965
974
10.4171/RMI/623
http://www.ems-ph.org/doi/10.4171/RMI/623
Loewner chains in the unit disk
Manuel
Contreras
Universidad de Sevilla, SEVILLA, SPAIN
Santiago
Díaz-Madrigal
Universidad de Sevilla, SEVILLA, SPAIN
Pavel
Gumenyuk
University of Bergen, BERGEN, NORWAY
Loewner chains, evolution families
In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.] of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.
Functions of a complex variable
Ordinary differential equations
General
975
1012
10.4171/RMI/624
http://www.ems-ph.org/doi/10.4171/RMI/624
Elliptic equations in the plane satisfying a Carleson measure condition
Martin
Dindoš
Edinburgh University, EDINBURGH, UNITED KINGDOM
David
Rule
Heriot-Watt University, EDINBURGH, UNITED KINGDOM
Elliptic equations, Carleson measure condition, Neumann problem, regularity problem, distributional inequalities, inhomogeneous equation
In this paper we settle (in dimension $n=2$) the open question whether for a divergence form equation $\div (A\nabla u) = 0$ with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the $L^p$ Neumann and Dirichlet regularity problems are solvable for some values of $p\in (1,\infty)$. The related question for the $L^p$ Dirichlet problem was settled (in any dimension) in 2001 by Kenig and Pipher [Kenig, C.E. and Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (2001), no. 1, 199-217].
Partial differential equations
General
1013
1034
10.4171/RMI/625
http://www.ems-ph.org/doi/10.4171/RMI/625
A counterexample for the geometric traveling salesman problem in the Heisenberg group
Nicolas
Juillet
Université de Grenoble I, SAINT-MARTIN D'HERES, FRANCE
Heisenberg group, Carnot-Carathéodory metric, rectifiable curve, Traveling Salesman Problem
We are interested in characterizing the compact sets of the Heisenberg group that are contained in a curve of finite length. Ferrari, Franchi and Pajot recently gave a sufficient condition for those sets, adapting a necessary and sufficient condition due to P. Jones in the Euclidean setting. We prove that this condition is not necessary.
Measure and integration
General
1035
1056
10.4171/RMI/626
http://www.ems-ph.org/doi/10.4171/RMI/626
Maps from Riemannian manifolds into non-degenerate Euclidean cones
Luciano
Mari
Università di Milano, MILANO, ITALY
Marco
Rigoli
Università di Milano, MILANO, ITALY
Maximum principles, harmonic maps, isometric immersion, Riemannian manifold
Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $\varphi: M \rightarrow \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $\varphi$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $\varphi$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $\varphi(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.
Differential geometry
Partial differential equations
General
1057
1074
10.4171/RMI/627
http://www.ems-ph.org/doi/10.4171/RMI/627
The $C^m$ Norm of a Function with Prescribed Jets I
Charles
Fefferman
Princeton University, PRINCETON, UNITED STATES
Whitney extension theorem, optimal $C^m$ norm
We prove a variant of the classical Whitney extension theorem, in which the $C^m$-norm of the extending function is controlled up to a given, small percentage error.
Calculus of variations and optimal control; optimization
Convex and discrete geometry
General
1075
1098
10.4171/RMI/628
http://www.ems-ph.org/doi/10.4171/RMI/628