Journal: Archive for Rational Mechanics and Analysis

Abbreviation

Arch Rational Mech Ana

Publisher

Springer

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ISSN

0003-9527
1432-0673

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Publications1 - 10 of 42
  • Degenerate Flat Bands in Twisted Bilayer Graphene
    Item type: Journal Article
    Becker , Simon; Humbert , Tristan; Zworski , Maciej (2026)
    Archive for Rational Mechanics and Analysis
    We prove that in the chiral limit of the Bistritzer–MacDonald Hamiltonian, there exist magic angles at which the Hamiltonian exhibits flat bands of multiplicity four instead of two. We analyse the structure of Bloch functions associated with the bands of arbitrary multiplicity, compute the corresponding Chern number to be -1, and show that there exist infinitely many degenerate magic angles for a generic choice of tunnelling potential, including the Bistritzer–MacDonald potential. Moreover, we demonstrate for generic tunnelling potentials that flat bands have only twofold or fourfold multiplicities.
  • Randomization and the Gross-Pitaevskii Hierarchy
    Item type: Journal Article
    Sohinger, Vedran; Staffilani, Gigliola (2015)
    Archive for Rational Mechanics and Analysis
    We study the Gross–Pitaevskii hierarchy on the spatial domain 𝕋�3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is 𝛼�>34 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7):4705–4764, 2014) that the range 𝛼�>1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1):169–185, 2008), where the spacetime estimate is known to hold whenever 𝛼�≥1 . The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross–Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross–Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.
  • The optimal partial transport problem
    Item type: Journal Article
    Figalli, Alessio (2010)
    Archive for Rational Mechanics and Analysis
  • A Resolution of the Poisson Problem for Elastic Plates
    Item type: Journal Article
    Da Lio, Francesca; Palmurella, Francesco; Rivière, Tristan (2020)
    Archive for Rational Mechanics and Analysis
  • A general class of free boundary problems for fully nonlinear elliptic equations
    Item type: Journal Article
    Figalli, Alessio; Shahgholian, Henrik (2014)
    Archive for Rational Mechanics and Analysis
  • Quantitative Characterization of Stress Concentration in the Presence of Closely Spaced Hard Inclusions in Two-Dimensional Linear Elasticity
    Item type: Journal Article
    Kang, Hyeonbae; Yu, Sanghyeon (2019)
    Archive for Rational Mechanics and Analysis
    In the region between close-to-touching hard inclusions, stress may be arbitrarily large as the inclusions get closer. This stress is represented by the gradient of a solution to the Lamé system of linear elasticity. We consider the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimating its magnitude. We introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lamé system, and then show that the singular behavior of the gradient in the narrow region can be precisely captured by singular functions. As a consequence of the characterization, we are able to regain the existing upper bound on the blow-up rate of the gradient, namely, ɛ⁻¹/² where ɛ is the distance between two inclusions. We then show that it is in fact an optimal bound by showing that there are cases where ɛ⁻¹/² is also a lower bound. This work is the first to completely reveal the singular nature of the gradient blow-up and to obtain the optimal blow-up rate in the context of the Lamé system with hard inclusions. The singular functions introduced in this paper play essential roles in overcoming the difficulties in applying the methods of previous works. The main tools of this paper are the layer potential techniques and the variational principle. The variational principle can be applied because the singular functions of this paper are solutions of the Lamé system.
  • On the Sharp Stability of Critical Points of the Sobolev Inequality
    Item type: Journal Article
    Figalli, Alessio; Glaudo, Federico (2020)
    Archive for Rational Mechanics and Analysis
    Given n≥ 3 , consider the critical elliptic equation Δu+u2∗-1=0 in Rn with u> 0. This equation corresponds to the Euler–Lagrange equation induced by the Sobolev embedding H1(Rn)↪L2∗(Rn), and it is well-known that the solutions are uniquely characterized and are given by the so-called “Talenti bubbles”. In addition, thanks to a fundamental result by Struwe (Math Z 187(4):511–517, 1984), this statement is “stable up to bubbling”: if u:Rn→(0,∞)almost solves Δu+u2∗-1=0 then u is (nonquantitatively) close in the H1(Rn) -norm to a sum of weakly-interacting Talenti bubbles. More precisely, if δ(u) denotes the H1(Rn) -distance of u from the manifold of sums of Talenti bubbles, Struwe proved that δ(u) → 0 as [InlineEquation not available: see fulltext.]. In this paper we investigate the validity of a sharp quantitative version of the stability for critical points: more precisely, we ask whether under a bound on the energy [InlineEquation not available: see fulltext.] (that controls the number of bubbles) it holds that [Equation not available: see fulltext.]A recent paper by the first author together with Ciraolo and Maggi (Int Math Res Not 2018(21):6780–6797, 2017) shows that the above result is true if u is close to only one bubble. Here we prove, to our surprise, that whenever there are at least two bubbles then the estimate above is true for 3 ≤ n≤ 5 while it is false for n≥ 6. To our knowledge, this is the first situation where quantitative stability estimates depend so strikingly on the dimension of the space, changing completely behavior for some particular value of the dimension n. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
  • Existence and uniqueness of maximal regular flows for non-smooth vector fields
    Item type: Journal Article
    Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio (2015)
    Archive for Rational Mechanics and Analysis
  • Mathematical Analysis of Plasmonic Nanoparticles: The Scalar Case
    Item type: Journal Article
    Ammari, Habib; Millien, Pierre; Ruiz, Matias; et al. (2017)
    Archive for Rational Mechanics and Analysis
    Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is fourfold: (1) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (2) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle; (3) to derive optimal bounds on the enhancement factors; (4) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.
  • Surface plasmon resonance of nanoparticles and applications in imaging
    Item type: Journal Article
    Ammari, Habib; Deng, Youjun; Millien, Pierre (2016)
    Archive for Rational Mechanics and Analysis
Publications1 - 10 of 42