Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \, dy. Here we consider a kernel $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$ where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on \rr^d. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the whole \rr^d and it is positive when the the determinant of the matrix $A$ is different from one. As an application of our results, we observe that, when the first eigenvalue is positive, there is an exponential decay for the solutions to the associated evolution problem. As a tool to obtain the result, we also study the behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the ball $B_R$ and prove that it converges to the first eigenvalue in the whole space as $R\to \infty$

Inverse problem for the heat equation and the Schrödinger equation on a tree

In this paper, we establish global Carleman estimates for the heat and Schrödinger equations on a network. The heat equation is considered on a general tree and the Schrödinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. Schrödinger) equation from boundary measurements

Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree k 2 for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points

Time-position characterization of conflicts: a case study of collaborative editing

International audienceCollaborative editing (CE) became increasingly common, often compulsory in academia and industry where people work in teams and are distributed across space and time. We aim to study collabora-tive editing behavior in terms of collaboration patterns users adopt and in terms of a characterisation of conflicts, i.e. edits from different users that occur close in time and position in the document. The process of a CE can be split into several editing 'sessions' which are performed by a single author ('single-authored session') or several authors ('co-authored session'). This fragmentation process requires a pre-defined 'maximum time gap' between sessions which is not yet well defined in previous studies. In this study, we analysed CE logs of 108 collaboratively edited documents. We show how to establish a suitable 'maximum time gap' to split CE activities into sessions by evaluating the distribution of the time distance between two adjacent sessions. We studied editing activities inside each 'co-author session' in order to define potential conflicts in terms of time and position dimensions before they occur in the document. We also analysed how many of these potential conflicts become real conflicts. Findings show that potential conflicting cases are few. However, they are more likely to become real conflicts

The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as 1 / \eps^2 we consider the asymptotic regime \eps \to 0 with the angular velocity $\Omega$ proportional to (\eps^2|\log\eps|)^{-1} . We prove that if \Omega = \Omega_0 (\eps^2|\log\eps|)^{-1} and $\Omega_0 > 2(3\pi)^{-1}$ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.Comment: 52 pages, PDFLaTex. Minor corrections, sign convention modified. To be published in Commun. Math. Phy