58 research outputs found

    Exchange Operator Formalism for Integrable Systems of Particles

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    We formulate one dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators. The hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level.Comment: 8 page

    A supercritical elliptic problem in a cylindrical shell

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    We consider the problem −Δu=∣u∣p−2uinΩ,u=0on∂Ω, -\Delta u=|u|^{p-2}u in \Omega, u=0 on \partial\Omega, where Ω:={(y,z)∈Rm+1×RN−m−1:0<a<∣y∣<b<∞}\Omega:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0<a<|y|<b<\infty\}, 0≀m≀N−10\leq m\leq N-1 and N≄2N\geq2. Let 2N,m∗:=2(N−m)/(N−m−2)2_{N,m}^{\ast}:=2(N-m)/(N-m-2) if m<N−2m<N-2 and 2N,m∗:=∞2_{N,m}^{\ast}:=\infty if m=N−2m=N-2 or N−1N-1. We show that 2N,m∗2_{N,m}^{\ast} is the true critical exponent for this problem, and that there exist nontrivial solutions if 2<p<2N,m∗2<p<2_{N,m}^{\ast} but there are no such solutions if p≄2N,m∗p\geq2_{N,m}^{\ast}

    A spinorial energy functional: critical points and gradient flow

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    On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, {\phi}) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor {\phi}. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.Comment: Small changes, final versio

    Four conjectures in Nonlinear Analysis

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    In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.Comment: arXiv admin note: text overlap with arXiv:1504.01010, arXiv:1409.5919, arXiv:1612.0819

    The Cauchy Problem for the Einstein Equations

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    Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in this context. The question of producing reduced systems of equations which are hyperbolic is examined in detail and some new results on that subject are presented. Relevant background from the theory of partial differential equations is also explained at some lengthComment: 98 page

    On the topology and area of higher dimensional black holes

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    Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher dimensional analogues of some well known results for black holes in 3+1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat (Λ=0\Lambda=0) black hole spacetimes, and Gibbons' and Woolgar's genus dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter (Λ<0\Lambda<0) spacetimes. In higher dimensions the genus is replaced by the so-called σ\sigma-constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.Comment: 15 pages, Latex2e; typos corrected, a convention clarified, resulting in the simplification of certain formulas, other improvement

    About curvature, conformal metrics and warped products

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    We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds (B,gB)(B,g_B) and (F,gF)(F,g_F) furnished with metrics of the form c2gB⊕w2gFc^{2}g_B \oplus w^2 g_F and, in particular, of the type w2ÎŒgB⊕w2gFw^{2 \mu}g_B \oplus w^2 g_F, where c,w ⁣:B→(0,∞)c, w \colon B \to (0,\infty) are smooth functions and ÎŒ\mu is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B,gB)(B,g_B) is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York type among others.Comment: 32 pages, 3 figure

    Magnetic vortex filament flows

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    We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrödinger equation showing the solitonic nature of those.Ministerio de Educación y CienciaFondo Europeo de Desarrollo RegionalJunta de Andalucí

    Small Horizons

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    All near horizon geometries of supersymmetric black holes in a N=2, D=5 higher-derivative supergravity theory are classified. Depending on the choice of near-horizon data we find that either there are no regular horizons, or horizons exist and the spatial cross-sections of the event horizons are conformal to a squashed or round S^3, S^1 * S^2, or T^3. If the conformal factor is constant then the solutions are maximally supersymmetric. If the conformal factor is not constant, we find that it satisfies a non-linear vortex equation, and the horizon may admit scalar hair.Comment: 21 pages, latex. Typos corrected and reference adde

    Local and global behaviour of nonlinear equations with natural growth terms

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    This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto
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