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European Mathematical Society Publishing House
2024-03-29 03:30:17
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=18&iss=9&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
9
Structure of classical (finite and affine) $\mathcal W$-algebras
Alberto
De Sole
Università di Roma La Sapienza, ROMA, ITALY
Victor
Kac
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Daniele
Valeri
SISSA, TRIESTE, ITALY
$\mathcal W$-algebra, Poisson algebra, Poisson vertex algebra, Slodowy slice, Hamiltonian reduction, Zhu algebra, Miura map
First, we derive an explicit formula for the Poisson bracket of the classical finite $\mathcal W$-algebra $\mathcal W^{\mathrm {fin}}(\mathfrak g,f)$, the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra $\mathfrak g$ and its nilpotent element $f$. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine $\mathcal W$-algebra $\mathcal W(\mathfrak g,f)$. As an immediate consequence, we obtain a Poisson algebra isomorphism between $\mathcal W^{\mathrm {fin}}(\mathfrak g,f)$ and the Zhu algebra of $\mathcal W(\mathfrak g,f)$. We also study the generalized Miura map for classical $\mathcal W$-algebras.
Nonassociative rings and algebras
Dynamical systems and ergodic theory
1873
1908
10.4171/JEMS/632
http://www.ems-ph.org/doi/10.4171/JEMS/632
The strong profinite genus of a finitely presented group can be infinite
Martin
Bridson
University of Oxford, OXFORD, UNITED KINGDOM
Profinite completion, profinite genus, Grothendieck pairs
We construct the first examples of finitely-presented, residually-finite groups $\Gamma$ that contain an infinite sequence of non-isomorphic finitely-presented subgroups $P_n \hookrightarrow \Gamma$ such that the inclusion maps induce isomorphisms of profinite completions $\widehat{P}_n \cong \widehat {\Gamma}$.
Group theory and generalizations
1909
1918
10.4171/JEMS/633
http://www.ems-ph.org/doi/10.4171/JEMS/633
Pseudo-holomorphic functions at the critical exponent
Laurent
Baratchart
INRIA, SOPHIA ANTIPOLIS CEDEX, FRANCE
Alexander
Borichev
Aix Marseille Université, MARSEILLE CEDEX 13, FRANCE
Slah
Chaabi
INRIA, SOPHIA ANTIPOLIS CEDEX, FRANCE
Pseudo-holomorphic functions, Hardy spaces, conjugate Beltrami equation, nonstrictly elliptic equations, Dirichlet problem
We study Hardy classes on the disk associated to the equation $\bar \partial w= \alpha \bar w$ for $\alpha \in L^r$ with $2 \leq r < \infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M. Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.
Functions of a complex variable
Partial differential equations
Functional analysis
1919
1960
10.4171/JEMS/634
http://www.ems-ph.org/doi/10.4171/JEMS/634
Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations
Marina
Ghisi
Università degli Studi di Pisa, PISA, ITALY
Massimo
Gobbino
Università di Pisa, PISA, ITALY
Alain
Haraux
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Semi-linear hyperbolic equation, dissipative hyperbolic equations, slow solutions, decay estimates, energy estimates
We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as $t \to \infty$, at least as fast as a suitable negative power of $t$. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of $t$. Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.
Partial differential equations
1961
1982
10.4171/JEMS/635
http://www.ems-ph.org/doi/10.4171/JEMS/635
Forking and JSJ decompositions in the free group
Chloé
Perin
Université de Strasbourg, STRASBOURG CEDEX, FRANCE
Rizos
Sklinos
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
Forking independence, torsion-free hyperbolic groups, curve complex
We give a description of the model-theoretic relation of forking independence in terms of the notion of JSJ decompositions in non-abelian free groups
Mathematical logic and foundations
Group theory and generalizations
1983
2017
10.4171/JEMS/636
http://www.ems-ph.org/doi/10.4171/JEMS/636
A minimization approach to hyperbolic Cauchy problems
Enrico
Serra
Politecnico di Torino, TORINO, ITALY
Paolo
Tilli
Politecnico di Torino, TORINO, ITALY
Nonlinear hyperbolic equations, mimimization, a priori estimates
Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of functions that minimize suitable functionals in space-time (where the initial data of the Cauchy Problem serve as prescribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the calculus of variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that fit in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.
Partial differential equations
Calculus of variations and optimal control; optimization
2019
2044
10.4171/JEMS/637
http://www.ems-ph.org/doi/10.4171/JEMS/637
Bounds on the Bondi energy by a flux of curvature
Spyros
Alexakis
University of Toronto, TORONTO, ONTARIO, CANADA
Arick
Shao
University of Toronto, TORONTO, ONTARIO, CANADA
Bondi, mass, energy, angular, momentum, curvature, flux, asymptotic, round, uniformization
We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy, momentum, and rate of energy loss. The bounds depend on the closeness between the given cone and a corresponding cone in a Schwarzschild spacetime, measured purely in terms of the differences between certain weighted $L^2$-norms of the spacetime curvature on the cones, and of the geometries of the spheres from which they emanate. This paper relies on the results in [1], which uniformly control the geometry of the given null cone up to infinity, as well as those of [18], which establish machinery for dealing with low regularities. A key step in this paper is the construction of a family of asymptotically round cuts of our cone, relative to which the Bondi energy is measured.
Relativity and gravitational theory
Partial differential equations
Differential geometry
2045
2106
10.4171/JEMS/638
http://www.ems-ph.org/doi/10.4171/JEMS/638
Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems
Alexander
Mielke
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Riccarda
Rossi
Università degli Studi di Brescia, BRESCIA, ITALY
Giuseppe
Savaré
Università di Pavia, PAVIA, ITALY
Rate-independent systems, energetic solutions, BV solutions, existence results, vanishing viscosity, time discretization
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower-semicontinuity compactness arguments, and on new BV-estimates that are of independent interest.
Partial differential equations
Ordinary differential equations
2107
2165
10.4171/JEMS/639
http://www.ems-ph.org/doi/10.4171/JEMS/639