- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 22:32:36
11
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=27&iss=1&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)
27
2011
1
Global existence for the primitive equations with small anisotropic viscosity
Frédéric
Charve
Université Paris 12 – Val de Marne, CRÉTEIL CEDEX, FRANCE
Van-Song
Ngo
Université Paris 12 – Val de Marne, CRÉTEIL CEDEX, FRANCE
Primitive equations, quasi-geostrophic system, anisotropy, dispersion, Strichartz estimates.
In this paper, we consider the primitive equations with zero vertical viscosity, zero vertical thermal diffusivity, and the horizontal viscosity and horizontal thermal diffusivity of size $\varepsilon^\alpha$ where $0 < \alpha < \alpha_0$. We prove the global existence of a unique strong solution for large data provided that the Rossby number is small enough (the rotation and the vertical stratification are large).
Partial differential equations
Fluid mechanics
General
1
38
10.4171/RMI/629
http://www.ems-ph.org/doi/10.4171/RMI/629
Le théorème du symbole total d’un opérateur différentiel p-adique d’échelon h ≥ 0
Zoghman
Mebkhout
Université Paris 7 Denis Diderot, PARIS CEDEX 05, FRANCE
p-adic differential operator, p-adic differential operator of h ≥ 0 echelon, total symbol, division, continuity, noetherianity, p-adic de Rham cohomology.
In this article we prove the total symbol theorem for the $p$-adic differential operators of degree $h\geq 0$ for the echelon filtration and the noetherianity of the ring of the $p$-adic differential operators of degree $h\geq 0$ for the echelon filtration over a $\dagger$-adic affine smooth scheme small enough.
Algebraic geometry
General
39
92
10.4171/RMI/630
http://www.ems-ph.org/doi/10.4171/RMI/630
Isoperimetry for spherically symmetric log-concave probability measures
Nolwen
Huet
Université de Toulouse, TOULOUSE, FRANCE
Isoperimetric inequalities, log-concave measures
We prove an isoperimetric inequality for probability measures $\mu$ on $\mathbb{R}^n$ with density proportional to $\exp(-\phi(\lambda |x|))$, where $|x|$ is the euclidean norm on $\mathbb{R}^n$ and $\phi$ is a non-decreasing convex function. It applies in particular when $\phi(x)=x^\alpha$ with $\alpha \ge 1$. Under mild assumptions on $\phi$, the inequality is dimension-free if $\lambda$ is chosen such that the covariance of $\mu$ is the identity.
Real functions
Measure and integration
Probability theory and stochastic processes
General
93
122
10.4171/RMI/631
http://www.ems-ph.org/doi/10.4171/RMI/631
Universal objects in categories of reproducing kernels
Daniel
Beltiţă
Romanian Academy, BUCHAREST, ROMANIA
José
Galé
Universidad de Zaragoza, ZARAGOZA, SPAIN
Reproducing kernel, category theory, vector bundle, tautological bundle, Grassmann manifold, completely positive map, universal object.
We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of $C^*$- algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing $(-*)$-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing $(-*)$-kernels with universality properties with respect to the operation of pull-back. We show how completely positive maps can be regarded as pull-backs of universal ones linked to the tautological bundle over the Grassmann manifold of the Hilbert space $\ell^2(\mathbb{N})$.
Functional analysis
Category theory; homological algebra
Operator theory
Global analysis, analysis on manifolds
123
179
10.4171/RMI/632
http://www.ems-ph.org/doi/10.4171/RMI/632
Tropical plane geometric constructions: a transfer technique in Tropical Geometry
Luis Felipe
Tabera Alonso
Universidad de Cantabria, SANTANDER, SPAIN
Tropical geometry, geometric constructions, incidence configurations.
The notion of geometric construction is introduced. This notion allows to compare incidence configurations both lying in the algebraic and the tropical plane. We provide sufficient conditions in a geometric construction to ensure that there is always an algebraic counterpart related by tropicalization. We also present some results to detect if this algebraic counterpart cannot exist. With these tools, geometric constructions are applied to transfer classical theorems to the tropical framework, we provide a notion of "constructible incidence theorem" and then several tropical versions of classical theorems are proved such as the converse of Pascal's, Fano's or Cayley-Bacharach theorems.
Algebraic geometry
Field theory and polynomials
Associative rings and algebras
Geometry
181
232
10.4171/RMI/633
http://www.ems-ph.org/doi/10.4171/RMI/633
Regularity for solutions of the total variation denoising problem
Vicent
Caselles
Universitat Pompeu-Fabra, BARCELONA, SPAIN
Antonin
Chambolle
Ecole Polytechnique, CNRS, PALAISEAU, FRANCE
Matteo
Novaga
Università di Pisa, PISA, ITALY
Image processing, variational methods, regularity of solutions.
The main purpose of this paper is to prove a local Hölder regularity result for the solutions of the total variation based denoising problem assuming that the datum is locally Hölder continuous. We also prove a global estimate on the modulus of continuity of the solution in convex domains of $\mathbb{R}^N$ and some extensions of this result for the total variation minimization flow.
Information and communication, circuits
Partial differential equations
Calculus of variations and optimal control; optimization
General
233
252
10.4171/RMI/634
http://www.ems-ph.org/doi/10.4171/RMI/634
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential
David
Ruiz
Universidad de Granada, GRANADA, SPAIN
Giusi
Vaira
SISSA, TRIESTE, ITALY
Nonlinear analysis, Schrödinger-Poisson-Slater problem, variational methods, singular perturbation method, multi-bump solutions.
In this paper we consider the system in $\mathbb{R}^3$ \begin{equation} \left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right. \end{equation} for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x)$. We point out that such solutions do not exist in the framework of the usual Nonlinear Schrödinger Equation.
Partial differential equations
General
253
271
10.4171/RMI/635
http://www.ems-ph.org/doi/10.4171/RMI/635
Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations
Raphaël
Côte
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
Yvan
Martel
École Polytechnique, PALAISEAU CEDEX, FRANCE
Frank
Merle
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Multi-solitons, generalized Korteweg-de Vries equation, nonlinear Schrödinger equation, instability, supercritical problem.
Multi-soliton solutions, i.e. solutions behaving as the sum of $N$ given solitons as $t \to +\infty$, were constructed for the $L^2$ critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140] and [Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864]). In this paper, we extend the construction of multi-soliton solutions to the $L^2$ supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
Partial differential equations
General
273
302
10.4171/RMI/636
http://www.ems-ph.org/doi/10.4171/RMI/636
Constant curvature foliations in asymptotically hyperbolic spaces
Rafe
Mazzeo
Stanford University, STANFORD, UNITED STATES
Frank
Pacard
École Polytechnique, PALAISEAU, FRANCE
Constant mean curvature, foliations, constant scalar curvature, Schouten tensor.
Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\partial M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant $\sigma_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for $g$ and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.
Differential geometry
General
303
333
10.4171/RMI/637
http://www.ems-ph.org/doi/10.4171/RMI/637
Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces
Outi Elena
Kansanen
KTH, STOCKHOLM, SWEDEN
Riikka
Korte
University of Helsinki, UNIVERSITY OF HELSINKI, FINLAND
Metric doubling measure, metric spaces, Muckenhoupt weights, strong $A_\infty$-weight.
We prove that every strong $A_\infty$-weight is a Muckenhoupt weight in Ahlfors-regular metric measure spaces that support a Poincaré inequality. We also explore the relations between various definitions for $A_\infty$-weights in this setting, since some of these characterizations are needed in the proof of the main result.
Fourier analysis
General
335
354
10.4171/RMI/638
http://www.ems-ph.org/doi/10.4171/RMI/638
The Jet of an Interpolant on a Finite Set
Charles
Fefferman
Princeton University, PRINCETON, UNITED STATES
Arie
Israel
University of Texas at Austin, AUSTIN, UNITED STATES
Interpolation, jet, algorithm, Whitney extension theorem.
We study functions $F \in C^m (\mathbb{R}^n)$ having norm less than a given constant $M$, and agreeing with a given function $f$ on a finite set $E$. Let $\Gamma_f (S,M)$ denote the convex set formed by taking the $(m-1)$-jets of all such $F$ at a given finite set $S \subset \mathbb{R}^n$. We provide an efficient algorithm to compute a convex polyhedron $\tilde{\Gamma}_f (S,M)$, such that $$ \Gamma_f (S,cM) \subset \tilde{\Gamma}_f (S,M) \subset \Gamma_f (S,CM), $ where $c$ and $C$ depend only on $m$ and $n$.
Calculus of variations and optimal control; optimization
Convex and discrete geometry
General
355
360
10.4171/RMI/639
http://www.ems-ph.org/doi/10.4171/RMI/639