- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 12:08:15
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=17&iss=6&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
17
2015
6
Regularity of Lipschitz free boundaries for the thin one-phase problem
Daniela
De Silva
Columbia University, NEW YORK, UNITED STATES
Ovidiu
Savin
Columbia University, NEW YORK, UNITED STATES
Energy minimizers, one-phase free boundary problem, monotonicity formula
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad \Omega \subset \mathbb R^{n+1},$$ among all functions $u\ge 0$ which are fixed on $\partial \Omega$.
Partial differential equations
1293
1326
10.4171/JEMS/531
http://www.ems-ph.org/doi/10.4171/JEMS/531
Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime
Hoai-Minh
Nguyen
EPFL SB CAMA, LAUSANNE, SWITZERLAND
Cloaking, anomalous localized resonance, negative index materials, complementary media
This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance and the blow up of the power of the fields as the loss of the material goes to 0. An important class of negative index materials for which the localized resonance might appear is the class of reflecting complementary media introduced in [24]. It was shown in [29] that the complementarity property is not enough to ensure a connection between the blow up of the power and the localized resonance. In this paper, we study CALR for a subclass of complementary media called doubly complementary media. This class is rich enough to allow us to cloak an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 placed in an arbitrary medium via anomalous localized resonance; the cloak is independent of the source. The following three properties are established for doubly complementary media: P1. CALR appears if and only if the power blows up; P2. The power blows up if the source is located "near” the plasmonic structure; P3. The power remains bounded if the source is far away from the plasmonic structure. Property P2, the blow up of the power, is in fact established for reflecting complementary media. The proofs are based on several new observations and ideas. One of the difficulties is to handle the localized resonance. To this end, we extend the reflecting and removing localized singularity techniques introduced in [24–26], and implement the separation of variables for Cauchy problems for a general shell. The results in this paper are inspired by and imply recent ones of Ammari et al. [3] and Kohn et al. [16] in two dimensions and extend theirs to general non-radial core-shell structures in both two and three dimensions.
Partial differential equations
Optics, electromagnetic theory
1327
1365
10.4171/JEMS/532
http://www.ems-ph.org/doi/10.4171/JEMS/532
Expansion in finite simple groups of Lie type
Emmanuel
Breuillard
Université Paris-Sud 11, ORSAY CEDEX, FRANCE
Ben
Green
University of Oxford, OXFORD, UNITED KINGDOM
Robert
Guralnick
University of Southern California, LOS ANGELES, UNITED STATES
Terence
Tao
University of California Los Angeles, LOS ANGELES, UNITED STATES
Finite simple groups, expander graphs, product theorems
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].
Group theory and generalizations
1367
1434
10.4171/JEMS/533
http://www.ems-ph.org/doi/10.4171/JEMS/533
Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows
Artur
Avila
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Marcelo
Viana
, RIO DE JANEIRO, BRAZIL
Amie
Wilkinson
University of Chicago, CHICAGO, UNITED STATES
Lyapunov exponent, geodesic flow, partial hyperbolicity, disintegration, absolute continuity, rigidity
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
Dynamical systems and ergodic theory
1435
1462
10.4171/JEMS/534
http://www.ems-ph.org/doi/10.4171/JEMS/534
A variational analysis of a gauged nonlinear Schrödinger equation
Alessio
Pomponio
Politecnico di Bari, BARI, ITALY
David
Ruiz
Universidad de Granada, GRANADA, SPAIN
Gauged Schrödinger equations, Chern-Simons theory, variational methods, concentration compactness
This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - \Delta u(x) + \left( \omega + \frac{h^2(|x|)}{|x|^2} + \int_{|x|}^{+\infty} \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x),$$ where $$ h(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, ds.$$ This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in(1,3)$, the functional may be bounded from below or not, depending on $\omega $. Quite surprisingly, the threshold value for $\omega $ is explicit. From this study we prove existence and non-existence of positive solutions.
Partial differential equations
1463
1486
10.4171/JEMS/535
http://www.ems-ph.org/doi/10.4171/JEMS/535
On the motion of a curve by its binormal curvature
Robert
Jerrard
University of Toronto, TORONTO, ONTARIO, CANADA
Didier
Smets
UPMC, Université Paris 06,, PARIS CEDEX 05, FRANCE
Binormal curvature flow, integral current, oriented varifold
We propose a weak formulation for the binormal curvature flow of curves in $\mathbb R^3$. This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.
Differential geometry
Fluid mechanics
1487
1515
10.4171/JEMS/536
http://www.ems-ph.org/doi/10.4171/JEMS/536
GCD sums from Poisson integrals and systems of dilated functions
Christoph
Aistleitner
Technische Universität Graz, GRAZ, AUSTRIA
István
Berkes
Technische Universität Graz, GRAZ, AUSTRIA
Kristian
Seip
University of Trondheim, TRONDHEIM, NORWAY
GCD sums and matrices, Carleson–Hunt inequality, Poisson integral, polydisc, spectral norm, convergence of series of dilated functions
Upper bounds for GCD sums of the form $$\sum_{k,{\ell}=1}^N\frac{(\mathrm {gcd}(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}$$ are established, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0
Number theory
General
1517
1546
10.4171/JEMS/537
http://www.ems-ph.org/doi/10.4171/JEMS/537